r 


REESE  LIBRARY 


UNIVERSITY  OF  CALIFORNIA 

^eceived  _'  ,190     >. 

*  •  ,    •  •    ,  ^        -  * 

Accession  No.  .  ...8.2995     •.  Class  No.  '    . 


A  COURSE   IN 


MECHANICAL  DRAWING. 


BY 

JOHN     S.    REID, 

Instructor  in  Mechanical  Drawing  and  Designing^ 

Sibley  College,   Cornell  University, 

Ithaca,  N.    Y. 


SECOND   EDITION,    REVISED. 
FIRST    THOUSAND. 


NEW  YORK. 

JOHN   WILEY   &    SONS. 

LONDON  :   CHAPMAN  &  HALL,  LIMITED. 

1900. 


/ 

A 


Copyright,  1898, 

BY 

JOHN    S.    REID. 


SOBERT  DRUMMOND,   ELECTROTYPHR   AND   PRINTER,   NEW  YORK. 


PREFACE. 


IN  the  course  of  a  large  experience  as  an  instructor  in 
drawing  and  designing,  the  author  of  this  work  has  often  been 
called  upon  to  teach  the  elements  of  mechanical  drawing  to 
students  in  marine,  electrical,  railway,  and  mechanical  engi- 
neering. Having  tried  and  failed  to  find  a  book  on  the  sub- 
ject that  was  entirely  suitable  for  his  use  as  a  text-book,  he 
has  found  it  necessary  to  prepare  the  present  work. 

This  course  contains,  in  the  author's  judgment,  a  com- 
plete and  concise  statement,  accompanied  by  examples,  of 
the  essential  principles  of  mechanical  drawing — all  that  any 
young  man  of  ordinary  intelligence  needs  to  master,  by  care- 
ful study,  the  more  advanced  problems  met  with  in  machine 
construction  and  design.  Such  works  as  the  author  has  tried, 
although  most  excellent  from  certain  standpoints,  were  either 
incomplete  in  some  of  the  divisions  of  the  subject  or  too  volu- 
minous and  elementary  in  the  treatment  of  details. 

The  author  does  not  imagine  this  work  is  perfect,  but  he 
believes  that  it  comes  nearer  what  is  needed  in  teaching  the 
elements  of  mechanical  drawing  in  technical  schools,  high 
schools,  evening  drawing  schools,  and  colleges  than  any  work 
he  has  examined. 

The  chapter  on  Conventions  will  be  appreciated  by  students 


82995 


PREFA  CE. 

when  called  upon  to  execute  working  drawings  in  practical 
work.  The  methods  described  are  considered  by  the  author 
to  be  those  which  have  met  with  general  approval  by  the 
experienced  American  draftsmen  of  the  present  time. 

My  acknowledgments  are  due  to   E.  C.  Cleaves,  professor 
of   drawing,  Sibley  College,   Cornell  University,   for  reading 

the  manuscript  and  making  some  valuable  suggestions. 

THE  AUTHOR. 

April  i,  1898. 


CONTENTS. 


INTRODUCTION. 

PAGE 

THE  COMPLETE  OUTFIT,  ILLUSTRATED  ...............................  i 

CHAPTER   I. 

INSTRUMENTS  .......................................................  7 

Use  of  Instruments  .............................................  7 

Pencil  ..........................................................  7 

Drawing  Pen  ...................................................  g 

Triangles  .......................................................  !  ! 

T  Square  .......................................................  !  r 

Drawing  Board  .................................................  ri 

Sibley  College  Scale  ............................................  I2 

Scale  Guard  ....................................................  I2 

Compasses  .....................................................  j-j 

Dividers  or  Spacers  .............................................  t-j 

Spring  Bows  ....................................................  r^ 

Irregular  Curves  ................................................  x^ 

Protractor  ......................................................  H 

CHAPTER    II. 
GEOMETRICAL  DRAWING  ............................................. 


CHAPTER   III. 
CONVENTIONS  .................................  ;  .....................     56 

CHAPTER   IV. 
LETTERING  AND  FIGURING  ...........................................     64 

CHAPTER   V. 
ORTHOGRAPHIC  PROJECTION  .........................................     74 

Shade  Lines,  Shades,  and  Shadows  ..............................    103 

Conventions  ..............................................  IO4 

Shades  ....................  .....................................   IO5 

Shadows  .................................................  e   U! 

Isometrical  Drawing  ............................................   I22 

Working  Drawings  .............................................  I2g 


MECHANICAL   DRAWING. 


INTRODUCTION. 

A  NEED  has  been  felt  by  instructors  and  students,  especially 
in  technical  courses,  for  a  text-book  that  would  illustrate  the 
fundamental  principles  of  mechanical  drawing  in  such  a  prac- 
tical, lucid,  direct  and  progressive  way  as  to  enable  the 
instructor  to  teach,  and  the  student  to  acquire,  the  greatest 
number  of  the  essential  principles  involved,  and  the  ability  to 
apply  them,  in  a  draftsman-like  manner,  in  the  shortest  space 
of  time. 

With  this  in  mind,  the  present  work  has  been  prepared 
from  the  experience  of  the  writer,  a  practical  draftsman  and 
teacher  for  over  fifteen  years. 

THE   COMPLETE   OUTFIT. 

The  complete  outfit  for  students  in  mechanical  drawing  in 
Sibley  College  is  as  follows : 

(i)  THE  DRAWING-BOARD  for  freshman  work  is  if  x  22" 
X  -f",  the  same  as  that  used  for  free-hand  drawing.  The 
board  for  sophomore  and  junior  drawing  is  20"  X  26"  X  not 
more  than  J"  in  thickness.  The  material  should  be  soft  pine 
and  constructed  as  shown  by  Fig.  I. 


MECHANICAL   DRAWING. 


(2)  PAPER,  quality  and  size  to  suit. 

(3)  PENCILS,  one  6H  and  one  4H   Koh-i-noor  or  Faber, 
also  one  Eagle  Pilot  No.  2  with  rubber  tip. 

(4)  The  T-SQUARE  for  freshman  work  is  furnished  by  the 


department ;  a  plain  pearwood  T-square  with  a  fixed  head  is 
all  that  is  necessary  for  sophomore  or  junior  work.  Length 
to  suit  drawing-board. 

(5)  INSTRUMENTS.  "The  Siblcy  College  Set,"  shown  by 
Fig.  2,  was  compiled  by  the  writer,  and  is  recommended  as  a 
first-class  medium-priced  set  of  instruments.  It  contains* 


FIG.  2. 

A  COMPASS,  5^"  long,  with  fixed  needle-point,  pencil,  pen 
and  lengthening  bar;  a  SPRING  Bow  PENCIL,  3"  long;  a 
SPRING  Bow  PEN,  3"  long;  a  SPRING  Bow  SPACER,  3"  long; 
a  DRAWING-PEN,  medium  length ;  a  HAIR-SPRING  DIVIDER, 
5" Jong;  a  nickel-plated  box  with  leads. 

(6)  A  TRIANGULAR  BOXWOOD  SCALE  graduated  as  fol- 
lows: 4//and2//,  3"  and  ij",  i",andi",  f'and  f",  TV  and  ^" . 


INTRODUCTION. 


FIG.  3. 


FIG.  4. 


FIG.  5 


MECHANICAL    DRA  WING. 


(7)  i  TRIANGLE  30°  x  60°,  celluloid,  10"  long.     Fig.  4. 


45°,  "          7" 


•(8)  "SiBLEY  COLLEGE  SET"  of  IRREGULAR  CURVES. 
(9)  GLASS-PAPER  PENCIL  SHARPENER. 


FIG.  6. 


(10)  INK,  black  waterproof,  S.&H.      Fig.  7. 
(n)      "      red  "  Higgins.      Fig.  8. 

(12)      "      blue 


FIG.  7. 


FIG.  8. 


(13)  INK  ERASER,  Faber's  Typewriter. 

(14)  PENCIL  ERASER,  Tower's  Multiplex  Rubber.    Fig.  9. 


IN  TROD  UCTJON. 


s 


(15)  SPONGE   RUBBER  or  FABER'S  KNEADED  RUBBER. 
Fig.   10. 


FIG.  9. 

(16)  TACKS,  a  small  box  of  i  oz.  tacks. 

(17)  WATER-COLORS,  £  pan  each  of  Payne's  Gray,  Crim- 
son Lake,  Prussian  Blue,  Burnt  Sienna,  and  Gamboge.    Wind- 
sor &  Newton.      Fig.  n. 


FIG.  10.  FIG.  ii. 

(18)  TINTING  BRUSH,  Camel's  Hair  No.  10.     Fig.  12. 


FIG.  12. 

(19)  TINTING  SAUCER.     Fig.  13. 

(20)  WATER  GLASS.     Fig.  14. 

(21)  ARKANSAS  OIL-STONE.     2"x  i" 

(22)  PIECE  OF  SHEET  CELLULOID  No.  300,  dull  on  both 
sides.      Thickness  TTJVir- 


MECHANICAL   DRA  WING. 


(23)  PROTRACTOR,  German  silver,  about  5"diam.    Fig.  15, 

(24)  SCALE  GUARD,     " 


FIG.   13. 


FIG.   14. 


(25)  SHEET  OF  TRACING-CLOTH,  18"  x  24". 

(26)  WRITING-PEN,  point,  "Gillott"  No.  303. 


FIG.  15. 


FIG.   16. 


(27)  Piece  of  SHEET  BRASS, 

(28)  NEEDLES,-  two  with  handles. 

The  following  numbers  of  "  The  Complete  Outfit  "  are 
all  that  the  student  will  be  required  to  purchase  for  freshman 
mechanical  drawing  (No.  2  Register,  '97^98):  2,  3,  5,  6,  7, 
8,  9,  10,  13,  14,  16,  26. 

The  remainder  of  the  outfit  may  be  purchased  during  the 
sophomore  and  junior  years. 


CHAPTER    I. 
INSTRUMENTS. 

IT  is  a  common  belief  among  students  that  any  kind  of 
cheap  instrument  will  do  with  which  to  learn  mechanical 
drawing,  and  not  until  they  have  acquired  the  proper  use  of 
the  instruments  should  they  spend  money  in  buying  a  first- 
class  set.  This  is  one  of  the  greatest  mistakes  that  can  be 
made.  Many  a  student  has  been  discouraged  and  disgusted 
because,  try  as  he  would,  he  could  not  make  a  good  drawing, 
using  a  set  of  instruments  with  which  it  would  be  difficult  for 
even  an  experienced  draftsman  to  make  a  creditable  showing. 

If  it  is  necessary  to  economize  in  this  direction  it  is  better 
and  easier  to  get  along  with  a  fewer  number,  and  have  them 
of  the  best,  than  it  is  to  have  an  elaborate  outfit  of  question- 
able quality. 

The  instruments  composing  the  "Sibley  College  Set" 
are  made  by  T.  Alteneder  &  Sons,  and  are  certainly  as  good 
as  the  best.  See  Fig.  17. 

USE    OF    INSTRUMENTS. 

The  Pencil. — Designs  of  all  kinds  are  usually  worked  out 
in  pencil  first,  and  if  to  be  finished  and  kept  they  are  inked  in 
and  sometimes  colored  and  shaded ;  but  if  the  drawing  is  only 
to  be  finished  in  pencil,  then  all  the  lines  except  construction, 
center,  and  dimension  lines  should  be  made  broad  and  dark, 

7 


8  MECHANICAL   DRAWING. 

so  that  the  drawing  will  stand  out  clear  and  distinct.  It  will 
be  noticed  that  this  calls  for  two  kinds  of  pencil-lines,  the 
first  a  thin,  even  line  made  with  a  hard,  fine-grained  lead- 
pencil,  not  less  than  6H  (either  Koh-i-noor  or  Faber's),  and 
sharpened  to  a  knife-edge  in  the  following  manner:  The  lead 
should  be  carefully  bared  of  the  wood  with  a  knife  for  about 
J",  and  the  wood  neatly  tapered  back  from  that  point ;  then 
lay  the  lead  upon  the  glass-paper  sharpener  illustrated  in  the 
outfit,  and  carefully  rub  to  and  fro  until  the  pencil  assumes  a 
long  taper  from  the  wood  to  the  point ;  now  turn  it  over  and 
do  the  same  with  the  other  side,  using  toward  the  last  a 
slightly  oscillating  motion  on  both  sides  until  the  point  has 
assumed  a  sharp,  thin,  knife-edge  endwise  and  an  elliptical 
contour  the  other  way. 

This  point  should  then  be  polished  on  a  piece  of  scrap 
drawing-paper  until  the  rough  burr  left  by  the  glass-paper  is 
removed,  leaving  a  smooth,  keen,  ideal  pencil-point  for  draw- 
ing straight  lines. 

With  such  a  point  but  little  pressure  is  required  in  the 
hands  of  the  draftsman  to  draw  the  most  desirable  line,  one 
that  can  be  easily  erased  when  necessary  and  inked  in  to 
much  better  advantage  than  if  the  line  had  been  made  with  a 
blunt  point,  because,  when  the  pencil-point  is  blunt  the  incli- 
nation is  to  press  hard  upon  it  when  drawing  a  line.  This 
forms  a  groove  in  the  paper  which  makes  it  very  difficult  to 
draw  an  even  inked  line. 

The  second  kind  of  a  pencil-line  is  the  broad  line,  as 
explained  above ;  it  should  be  drawn  with  a  somewhat  softer 
pencil,  say  4.H,  and  a  thicker  point. 

All  lines  not  necessary  to  explain  the  drawing  should  be 


INSTRUMENTS,  9 

erased  before  inking  or  broadening  the  pencil-lines,  so  as  to 
make  a  minimum  of  erasing  and  cleaning  after  the  drawing  is 
finished. 

When  drawing  pencil-lines,  the  pencil  should  be  held  in  a 
plane  passing  through  the  edge  of  the  T-square  perpen- 
dicular to  the  plane  of  the  paper  and  making  an  angle  with 

• 
the  plane  of  the  paper  equal  to  about  60°. 

Lines  should  always  be  drawn  from  left  to  right.  A  soft 
conical-pointed  pencil  should  be  used  for  lettering,  figuring, 
and  all  free-hand  work. 

The  Draiving-pen. — The  best  form,  in  the  writer's  opinion, 
is  that  shown  in  Fig.  17.  The  spring  on  the  upper  blade 


FIG.   17. 

spreads  the  blades  sufficiently  apart  to  allow  for  thorough 
cleaning  and  sharpening.  The  hinged  blade  is  therefore 

V 

unnecessary.  The  pen  should  be  held  in  a  plane  passing 
through  the  edge  of  the  T-square  at  right  angles  to  the  plane 
of  the  paper,  and  making  an  angle  with  the  plane  of  the 
paper  ranging  from  60°  to  90°. 

The   best   of  drawing-pens  will  in  time  wear  dull  on  the 
point,  and   until  the  student  has  learned  from  a  competent 


OF  THK 

UNIVERSITT 


10  MECHANICAL  DRAWING. 

teacher  how  to  sharpen  his  pens  it  would  be  better  to  have 
them  sharpened  by  the  manufacturer. 

It  is  difficult  to  explain  the  method  of  sharpening  a  draw- 
ing-pen. 

If  one  blade  has  worn  shorter  than  the  other,  the  blades 
should  be  brought  together  by  means  of  the  thumb-screw,  and 
placing  the  pen  in  an  upright  position  draw  the  point  to  and 
fro  on  the  oil-stone  in  a  plane  perpendicular  to  it,  raising  and 
lowering  the  handle  of  the  pen  at  the  same  time,  to  give  the 
proper  curve  to  the  point.  The  Arkansas  oil-stones  (No.  21 
of  "  The  Complete  Outfit  ")  are  best  for  this  purpose. 

The  blades  should  next  be  opened  slightly,  and  holding 
the  pen  in  the  right  hand  in  a  nearly  horizontal  position,  place 
the  lower  blade  on  the  stone  and  move  it  quickly  to  and  fro, 
slightly  turning  the  pen  with  the  fingers  and  elevating  the 
handle  a  little  at  the  end  of  each  stroke.  Having  ground  the 
lower  blade  a  little,  turn  the  pen  completely  over  and  grind 
the  upper  blade  in  a  similar  manner  for  about  the  same  length 
of  time ;  then  clean  the  blades  and  examine  the  extreme 
points,  and  if  there  are  still  bright  spots  to  be  seen  continue 
the  grinding  until  they  entirely  disappear,  and  finish  the 
sharpening  by  polishing  on  a  piece  of  smooth  leather. 

The  blades  should  not  be  too  sharp,  or  they  will  cut  the 
paper.  The  grinding  should  be  continued  only  as  long  as  the 
bright  spots  show  on  the  points  of  the  blades. 

When  inking,  the  pen  should  be  held  in  about  the  same 
position  as  described  for  holding  the  pencil.  Many  drafts- 
men hold  the  pen  vertically.  The  position  may  be  varied 
with  good  results  as  the  pen  wears.  Lines  made  with  the 
pen  should  only  be  drawn  from  left  to  right. 


AYS  TR  UMEN  TS.  1 1 

THE   TRIANGLES. 

The  triangles  shown  at  Fig.  4  (in  "  The  Complete  Outfit  ") 
are  10"  and  7"  long  respectively,  and  are  made  of  transparent 
celluloid.  The  black  rubber  triangles  sometimes  used  are  but 
very  little  cheaper  (about  10  cents)  and  soon  become  dirty 
when  in  use;  the  rubber  is  brittle  and  more  easily  broken  than 
the  celluloid. 

Angles  of  15°,  75°,  30°,  45°,  60°,  and  90°  can  readily  be 
drawn  with  the  triangles  and  T-square.  Lines  parallel  to 
oblique  lines  on  the  drawing  can  be  drawn  with  the  triangles 
by  placing  the  edge  representing  the  height  of  one  of  them 
so  as  to  coincide  with  the  given  line,  then  place  the  edge  rep- 
resenting the  hypotenuse  of  the  other  against  the  corre- 
sponding edge  of  the  first,  and  by  sliding  the  upper  on  the 
lower  when  holding  the  lower  firmly  with  the  left  hand  any 
number  of  lines  may  be  drawn  parallel  to  the  given  line. 

The  methods  of  drawing  perpendicular  lines  and  making 
angles  with  other  lines  within  the  scope  of  the  triangles  and  T- 
square  are  so  evident  that  further  explanation  is  unnecessary. 

THE   T-SQUARE. 

The  use  of  the  T-square  is  very  simple,  and  is  accom- 
plished by  holding  the  head  firmly  with  the  left  hand  against 
the  left-hand  end  of  the  drawing-board,  leaving  the  right 
hand  free  to  use  the  pen  or  pencil  in  drawing  the  required 
lines. 

THE    DRAWING-BOARD. 

If  the  left-hand  edge  of  the  drawing-board  is  straight  and 
even  and  the  paper  is  tacked  down  square  with  that  edge  and 


12  MECHANICAL   DRAWING. 

the  T-square,  then  horizontal  lines  parallel  to  the  upper  edge 
of  the  paper  and  perpendicular  to  the  left-hand  edge  may  be 
drawn  with  the  T-square,  and  lines  perpendicular  to  these 
can  be  made  by  means  of  the  triangles,  or  set  squares,  as  they 
are  sometimes  called. 

THE    SIBLEY    COLLEGE    SCALE. 

This  scale,  illustrated  in  Fig.  3  (in  "  The  Complete  Out- 
fit "),  was  arranged  to  suit  the  needs  of  the  students  in  Sibley 
College.  It  is  triangular  and  made  of  boxwood.  The  six 
edges  are  graduated  as  follows;  TV'  or  full  size,  -g^",  J" 
and  |';  =  i  ft.,  i"  and  \"  =  I  ft.,  3"  and  ii"  =  I  ft.,  and 
4"  and  2"  —l  ft. 

Drawings  of  very  small  objects  are  generally  shown  en- 
larged— e.g.,  if  it  is  determined  to  make  a  drawing  twice  the 
full  size  of  an  object,  then  where  the  object  measures  one  inch 
the  drawing  would  be  made  2" ',  etc. 

Larger  objects  or  small  machine  parts  are  often  drawn  full 
size — i.e.,  the  same  size  as  the  object  really  is — and  the  draw- 
ing is  said  to  be  made  to  the  scale  of  full  size. 

Large  machines  and  large  details  are  usually  made  to  a 
reduced  scale — e.g.,  if  a  drawing  is  to  be  made  to  the  scale  of 
2"  •—  I  ft.,  then  2"  measured  by  the  standard  rule  would  be 
divided  into  12  equal  parts  and  each  part  would  represent  i" . 
See  Fig.  8i£. 

THE    SCALE    GUARD. 

This  instrument  is  shown  in  Fig.  16  (in  "The  Complete 
Outfit ").  It  is  employed  to  prevent  the  scale  from  turning, 
so  that  the  draftsman  can  use  it  without  having  to  look  for 


S.VS  TR  UMEN  TS.  1 3 

the  particular  edge  he  needs  every  time  he  wants  to  lay  off 
a  measurement. 

THE   COMPASSES. 

When  about  to  draw  a  circle  or  an  arc  of  a  circle,  take 
hold  of  the  compass  at  the  joint  with  the  thumb  and  two  first 
fingers,  guide  the  needle-point  into  the  center  and  set  the 
pencil  or  pen  leg  to  the  required  radrus,  then  move  the  thumb 
and  forefinger  up  to  the  small  handle  provided  at  the  top  of 
the  instrument,  and  beginning  at  the  lowest  point  draw  the 
line  clockwise.  The  weight  of  the  compass  will  be  the  only 
down  pressure  required. 

The  sharpening  of  the  lead  for  the  compasses  is  a  very  im- 
portant matter,  and  cannot  be  emphasized  too  much.  Before 
commencing  a  drawing  it  pays  well  to  take  time  to  properly 
sharpen  the  pencil  and  the  lead  for  compasses  and  to  keep 
them  always  in  good  condition. 

The  directions  for  sharpening  the  compass  leads  are  the 
same  as  has  already  been  given  for  the  sharpening  of  the 
straight-line  pencil. 

THE   DIVIDERS    OR    SPACERS. 

This  instrument  should  be  held  in  the  same  manner  as  de- 
scribed for  the  compass.  It  is  very  useful  in  laying  off  equal 
distances  on  straight  lines  or  circles.  To  divide  a  given  line 
into  any  number  of  equal  parts  with  the  dividers,  say  12,  it 
is  best  to  divide  the  line  into  three  or  four  parts  first,  say  4, 
and  then  when  one  of  these  parts  has  been  subdivided  accu- 
rately into  three  equal  parts,  it  will  be  a  simple  matter  to 
step  off  these  latter  divisions  on  the  remaining  three-fourths 


14  MECHANICAL  DRAWING. 

of  the  given  line.  Care  should  be  taken  not  to  make  holes  in 
the  paper  with  the  spacers,  as  it  is  difficult  to  ink  over  them 
without  blotting. 

THE    SPRING    BOWS. 

These  instruments  are  valuable  for  drawing  the  small  cir- 
cles and  arcs  of  circles.  It  is  very  important  that  all  the 
small  arcs,  such  as  fillets,  round  corners,  etc.,  should  be  care- 
fully pencilled  in  before  beginning  to  ink  a  drawing.  Many 
good  drawings  are  spoiled  because  of  the  bad  joints  between 
small  arcs  and  straight  lines. 

When  commencing  to  ink  a  drawing,  all  small  arcs  and 
small  circles  should  be  inked  first,  then  the  larger  arcs  and 
circles,  and  the  straight  lines  last.  This  is  best,  because  it  is 
much  easier  to  know  where  to  stop  the  arc  line,  and  to  draw 
the  straight  line  tangent  to  it,  than  vice  versa. 

IRREGULAR   CURVES. 

The  Sibley  College  Set  of  Irregular  Curves  shown  in  Fig. 
5  are  useful  for  drawing  irregular  curves  through  points  that 
have  already  been  found  by  construction,  such  as  ellipses, 
cycloids,  epicyloids,  etc.,  as  in  the  cases  of  gear-teeth,  cam 
outlines,  rotary  pump  wheels,  etc. 

When  using  these  curves,  that  curve  should  be  selected 
that  will  coincide  with  the  greatest  number  of  points  on  the 
line  required. 

THE    PROTRACTOR. 

This  instrument  is  for  measuring  and  constructing  angles. 
It  is  shown  in  Fig.  15.  It  is  used  as  follows  when  measuring 


INS  TR  UMEN  TS.  1 5 

an  angle:  Place  the  lower  straight  edge  on  the  straight  line 
which  forms  one  of  the  sides  of  the  angle,  with  the  nick 
exactly  on  the  point  of  the  angle  to  be  measured.  Then  the 
number  of  degrees  contained  in  the  angle  may  be  read  from 
the  left,  clockwise. 

In  constructing  an  angle,  place  the  nick  at  the  point  from 
which  it  is  desired  to  draw  the  angle,  and  on  the  outer  circum- 
ference of  the  protractor,  find  the  figure  corresponding  to  the 
nilmber  of  degrees  in  the  required  angle,  and  mark  a  point  on 
the  paper  as  close  as  possible  to  the  figure  on  the  protractor ; 
after  removing  the  protractor,  draw  a  line  through  this  point 
to  the  nick,  which  will  give  the  required  angle. 


CHAPTER  II. 
GEOMETRICAL  DRAWING. 

The  following  problems  are  given  to  serve  a  double  pur- 
pose :  to  teach  the  use  of  drawing  instruments,  and  to  point 
out  those  problems  in  practical  geometry  that  are  most  useful 
in  mechanical  drawing,  and  to  impress  them  upon  the  mind  of 
the  student  so  that  he  may  readily  apply  them  in  practice. 

The  drawing-paper  for  this  work  should  be  divided  tem- 
porarily, with  light  pencil-lines,  into  as  many  squares  and  rec- 
tangles as  may  be  directed  by  the  instructor,  and  the  drawings 
made  as  large  as  the  size  of  the  squares  will  permit.  The 
average  size  of  the  squares  should  be  not  less  than  4".  When 
a  sheet  of  drawings  is  finished  these  boundary  lines  may  be 
erased. 

It  will  be  noticed  in  the  illustrations  of  this  chapter  that 
all  construction  lines  are  made  very  narrow,  and  given  and 
required  lines  quite  broad.  This  is  sufficient  to  distinguish 
them,  and  employs  less  time  than  would  be  necessary  if  the 
construction  lines  were  made  broken,  as  is  often  the  case. 

If  time  will  permit,  it  is  advisable  to  ink  in  some  of  these 
drawings  toward  the  last.  In  that  event,  the  given  lines  may 
be  red,  the  construction  lines  blue,  and  the  required  lines 
black. 

But  even  when  inked  in  in  black,  the  broad  and  narrow 

16 


GEOMETRICAL   DRAWING.  I/ 

lines  would  serve  the  purpose  very  well  without  the  use  of  col- 
ored inks. 

The  principal  thing  to  be  aimed  at  in  making  these  draw- 
ings is  accuracy  of  construction.  All  dimensions  should  be 
laid  off  carefully,  correctly,  and  quickly.  Straight  lines  join- 
ing arcs  should  be  exactly  tangent,  so  that  the  joints  cannot 
be  noticed.  It  is  the  little  things  like  these  that  make  or  mar 
a  drawing,  and  if  attended  to  or  neglected  they  will  make  or 
mar  the  draftsman.  The  constant  endeavor  of  the  student 
should  be  to  make  every  drawing  lie  begins  more  accurate, 
quicker  and  better  in  every  way  than  the  preceding  one. 

A  drawing  should  never  be  handed  in  as  finished  until  the 
student  is  perfectly  sure  that  he  cannot  improve  it  in  any  way 
whatever,  for  the  act  of  handing  in  a  drawing  is  the  same,  or 
should  be  the  same,  as  saying  This  is  the  best  that  I  can  do ; 
I  cannot  improve  it ;  it  is  a  true  measure  of  my  ability  to 
make  this  drawing.  ,-'" 

If  these  suggestions  are  faithfully  followed  throughout  this 
course,  success  awaits  any  one  who  earnestly  desires  it. 

FIG.  1 8.  To  BISECT  A  FINITE  STRAIGHT  LINE. — With 
A  and  B  in  turn'  as  centers,  and  a  radius  greater  than  the  half 
of  AB,  draw  arcs  intersecting  at  E  and  F.  Join  EF  bisect- 
ing AB  at  C. 

An  arc  of. a  circle  may  be  bisected  in  the  same  way. 

FIG.  19.  To  ERECT  A  PERPENDICULAR  AT  THE  END  OF 
THE  LINE. — Assume  the  point  E  above  the  line  as  center  and 
radius  EB  describe  an  arc  CBD  cutting  the  line  AB  in  the 
point  C.  From  C  draw  a  line  through  E  cutting  the  arc  in 
D.  Draw  DB  the  perpendicular. 

FIG.  20.  THE  SAME  PROBLEM:  A  SECOND  METHOD. — 


18 


MECHANICAL   DRAWING. 


With  center  B  and  any  radius  as  BC  describe  an  arc  CDE 
with  the  same  radius;  measure  off  the  arcs  CDzndDE.  With 
C  and  D  as  centers  and  any  convenient  radius  describe  arcs  in- 
tersecting at  F.  FB  is  the  required  perpendicular. 


FIG.  21. 

FIG.  21.  To  DRAW  A  PERPENDICULAR  TO  A  LINE 
FROM  A  POINT  ABOVE  OR  BELOW  IT. — Assume  the  point 
C  above  the  line.  With  center  C  and  any  suitable  radius 
cut  the  line  AB  in  E  and  F.  From  E  and  F  describe  arcs- 
cutting  in  D.  Draw  CD  the  perpendicular  required. 


GEOMETRICAL  DRA  WING.  19 

FIG.  22.  To  BISECT  A  GIVEN  ANGLE. — With  A  as  center 
and  any  convenient  radius  describe  the  arc  BC.  With  B  and 
C  as  centers  and  any  convenient  radius  draw  arcs  intersecting 
at  D.  Join  AD,  then  angle  BAD  =  angle  DAC. 


FIG.  22. 

FIG.  23.  To  DRAW  A  LINE  PARALLEL  TO  A  GIVEN 
LINE  AB  THROUGH  A  GIVEN  POINT  C. — From  any  point 
on  AB  as  B  with  radius  BC  describe  an  arc  cutting  AB  in  A. 
From  C  with  the  same  radius  describe  arc  BD.  From  B  with 
AC  as  radius  cut  arc  BD  in  D.  Draw  CD.  Line  CD  is  paral- 
lel to  AB. 

z> 


FIG.  23. 

FIG.  24.  FROM  A  POINT  D  ON  THE  LINE  DE  TO  SET 
OFF  AN  ANGLE  EQUAL  TO  THE  GIVEN  ANGLE  BAC. — From 


20 


MECHANICAL   DRA  WING. 


A  with  any  convenient  radius  describe  arc  BC.  From  D  with 
the  same  radius  describe  arc  EF.  With  center  E  and  radius 
BC  cut  arc  EF  in  F.  Join  DF.  Angle  EDF  is  =  angle  BAG. 


D 


FIG.  24. 


FIG.  25.  To  DIVIDE  AN  ANGLE  INTO  TWO  EQUAL 
PARTS,  WHEN  THE  LINES  DO  NOT  EXTEND  TO  A  MEETING 
POINT.  —  Draw  the  line  CD  and  CE  parallel  and  at  equal  dis- 


FIG.  25. 


tances  from  the  lines  AB  and  FG.     With  C  as  center  and  any 
radius  draw  arcs  1,2.     With  i  and  2  as  centers  and  any  con- 


GEOMETRICAL   DRAWING.  21 

venient  radius  describe  arcs  intersecting  at  H.    A  line  through 
C  and  H  divides  the  angle  into  two  equal  parts. 

FIG.  26.  To  CONSTRUCT  A  RHOMBOID  HAVING  ADJA- 
CENT SIDES  EQUAL  TO  TWO  GIVEN  LINES  AB  AND  AC,  AND 
AN  ANGLE  EQUAL  TO  A  GIVEN  ANGLE  A. — Draw  line  DE 
equal  to  AB.  Make  D  —  angle  A.  Make  DF  —  AC.  From 
T^with  line  AB  as  radius  and  from  E  with  line  A  C  as  radius 
describe  arcs  cutting  \\\~G.  Join  FG  and  EG. 


FIG.  27.  To  DIVIDE  THE  LINE  AB  INTO  ANY  NUMBER 
OF  EQUAL  PARTS,  SAY  15. — Draw  a  line  CD  parallel  to  AB, 
of  any  convenient  length.  From  C  set  off  along  this  line  the 
number  of  equal  parts  into  which  the  \mzAB  is  to  be  divided. 
Draw  CA  and  DB  and  produce  them  until  they  intersect  at 
E.  Through  each  one  of  the  points  I,  2,  3,  4,  etc.,  draw 
lines  to  the  point  E,  dividing  the  line  AB  into  the  required 
number  of  equal  parts. 

This  problem  is  useful  in  dividing  a  line  when  the  point 
required  is  difficult  to  find  accurately — e.g.,  in  Fig.  28  AB  is 
the  pitch  of  the  spur  gear,  partly  shown,  which  includes  a 


22 


ME  CHA  NIC  A  L   DRAW  ING. 


space  and  a  tooth  and  is  measured  on  the  pitch  circle.  In 
cast  gears  the  space  is  made  larger  than  the  thickness  of  the 
tooth,  the  proportion  being  about  6  to  5 — i.e.,  if  we  divide 
the  pitch  into  eleven  equal  parts  the  space  will  measure  T6T 


S  4-   5  6    7  8   9  1011 12 13  U  If 
FIG.    27. 


FIG    28. 


and  the  tooth  T5T.  The  yV  which  the  space  is  larger  than  the 
tooth  is  called  the  backlash.  Let  A'B'  be  the  pitch  chord  of 
the  arc  AB.  Draw  CD  parallel  to  A'B'  at  any  convenient 
distance  and  set  off  on  it  I  \  equal  spaces  of  any  convenient 
length.  Draw  CAf  and  DB'  intersecting  at  E.  From  point 
5  draw  a  line  to  E  which  will  divide  A'B'  as  required ;  the 
one  part  T5T  and  the  other  T6T. 

FIG.  29.  To  DIVIDE  A  GIVEN  LINE  INTO  ANY  NUMBER 
OF  EQUAL  PARTS:  ANOTHER  METHOD. — Let  AB  be  the 
given  line.  From  A  draw  A  C  at  any  angle,  and  lay  off  on  it 
the  required  number  of  equal  spaces  of  any  convenient  length. 
Join  CB  and  through  the  divisions  on  AC  draw  lines  parallel 
to  CB,  dividing  AB  as  required  in  the  points  i',  2',  3',  4',  etc. 

FIG.  30.  To  DIVIDE  A  LINE  AB  PROPORTIONALLY  TO 
THE  DIVIDED  LINE  CD. — Draw  AB  parallel  to  CD  at  any 


GEOMETRICAL  DRAWING. 


distance  from  it.  Draw  lines  through  CA  and  DB  and  produce 
them  till  they  meet  at  E.  Draw  lines  from  E  through  the 
divisions  I,  ?,  3,  4,  etc.,  of  line  CD,  cutting  line  AB  in  the 


3456  7   8  9  10111213  14  B 
FIG.  2q. 


points  5,  6,  7,  8,  etc.  The  divisions  on  AB  will  have  the 
same  proportion  to  the  divisions  on  CD  that  the  whole  line 
AB  has  to  the  whole  line  CD  —  i.e.,  the  lines  will  be  propor- 
tionally divided. 


FIG.  31.    THE   SAME:    ANOTHER   METHOD. — Let  BC, 
the  divided  line,  make  any  angle  with  BA,  the  line  to  be  di- 


24  MECHANICAL   DRAWING. 

vided  at  B.  Draw  line  CA  joining  the  two  ends  of  the  lines. 
Draw  lines  from  5,  6,  7,  8,  parallel  to  CA,  dividing  line  AB 
in  points  I,  2,  3,  4,  proportional  to  BC. 

FIG.  32.  To  CONSTRUCT  AN  EQUILATERAL  TRIANGLE 
ON  A  GIVEN  BASE  AB. — From  the  points  A  and  B  with  AB 
as  radius  describe  arcs  cutting  in  C.  Draw  lines  AC  and  BC. 
The  triangle  ABC  is  equilateral  and  equiangular. 


FIG.  33.  To  CONSTRUCT  AN  EQUILATERAL  TRIANGLE 
OF  A  GIVEN  ALTITUDE,  AB. — From  both  ends  of  AB  draw 
lines  perpendicular  to  it  as  CA  and  DB.  From  A  with  any 
radius  describe  a  semicircle  on  CA  and  with  its  radius  cut  off 
arcs  I,  2.  Draw  lines  from  A  through  I,  2,  and  produce 
them  until  they  cut  the  base  BD. 

FIG.  34.  To  TRISECT  A  RIGHT  ANGLE  ABC. —  From 
the  angular  point  B  with  any  convenient  radius  describe  an 
arc  cutting  the  sides  of  the  angle  in  C  and  A.  From  C  and  A 
with  the  same  radius  cut  off  arcs  I  and  2.  Draw  lines  iB  and 
2B,  and  the  right  angle  will  be  trisected. 


GEOMETRICAL  DRAWING. 


FIG.  35.  To  CONSTRUCT  ANY  TRIANGLE,  ITS  THREE 
SIDES  AB  AND  C  BEING  GIVEN. — From  one  end  of  the  base 
as  A  describe  an  arc  with  the  line  B  as  radius.  From  the 
other  end  with  line  C  as  radius  describe  an  arc,  cutting  the 
first  arc  in  D.  From  D  draw  lines  to  the  ends  of  line  A,  and  a 
triangle  will  be  constructed  having  its  sides  equal  to  the  sides 
given.  To  construct  any  triangle  the  two  shorter  sides  B  and 
C  must  together  be  more  than  equal  to  the  largest  side  A. 


A 
'  B 


FIG.  34- 


FIG.  35- 


FIG.  36. 


'  FIG.  37- 


FIG.    36.   To   CONSTRUCT    A   SQUARE,   ITS  BASE  AB 
BEING  GIVEN. — Erect  a  perpendicular  at  B.     Make  BC  equal 


26 


MECHANICAL   DRAWING. 


to  AB.  From  A  and  C  with  radius  AB  describe  arcs  cutting 
in  D.  Join  DC  and  DA. 

FIG.  37.  To  CONSTRUCT  A  SQUARE,  GIVEN  ITS  DI- 
AGONAL AB. — Bisect  AB  in  C.  Draw  DF  perpendicular  to 
AB  at  C.  Make  CD  and  £F  each  equal  to  CA.  Join  ^4£>, 
/>£,  BF,  and  /^. 

FIG.  38.  To  CONSTRUCT  A  REGULAR  POLYGON  OF  ANY 
NUMBER  OF  SIDES,  THE  CIRCUMSCRIBING  CIRCLE  BEING 
GIVEN. — At  any  point  of  contact,  as  C,  draw  a  tangent  AB 
to  the  given  circle.  From  C  with  any  radius  describe  a  semi- 
circle cutting  the  given  circle.  Divide  the  semicircle  into  as 
many  equal  parts  as  the  polygon  is  required  to  have  sides,  as 
i,  2,  3,  4,  5,  6.^  Draw  lines  from  C  through  each  division, 
cutting  the  circle  in  points  which  will  give  the  angles  of  the 
polygon. 


FIG.  39.  ANOTHER  METHOD. — Draw  a  diameter  AB  of 
the  given  circle.  Divide  AB  into  as  many  equal  parts  as 
the  polygon  is  to  have  sides,  say  5.  From  A  and  B  with  the 


GEOMETRICAL   DRAWING. 


line  AB  as  radius  describe  arcs  cutting  in  C,  draw  a  line  from 
C  through  the  second  division  of  the  diameter  and  produce  it 
cutting  the  circle  in  D.  BD  will  be  the  side  of  the  required 
polygon.  The  line  C  must  always  be  drawn  through  the 
second  division  of  the  diameter,  whatever  the  number  of 
sides  of  the  polygon. 

FIG.  40.  To  CONSTRUCT  ANY  REGULAR  POLYGON 
WITH  A  GIVEN  SIDE  AB. — Make  BD  perpendicular  and 
equal  to  AB.  With  B  as  center  and  radius  AB  describe  arc 
DA.  Divide  arc  DA  into  as  many  equal  parts  as  there  are 
sides  in  the  required  polygon,  as  I,  2,  3,  4,  5.  Draw  B2. 
Bisect  line  AB  and  erect  a  perpendicular  at  the  bisection  cut- 
ting B2  in  C.  With  C  as  center  and  radius  CB  describe  a 
circle.  With  AB  as  a  chord  step  off  the  remaining  sides  of 
the  polygon. 


FIG.  40. 


FIG.  41. 


FIG.  41.  ANOTHER  METHOD. — Extend  Hne  AB.  With 
center  A  and  any  convenient  radius  describe  a  semicircle. 
Divide  the  semicircle  into  as  many  equal  parts  as  there  are 
sides  in  the  required  polygon,  say  6.  Draw  lines  through 
every  division  except  the  first.  With  A  as  center  and  AB  as 


28 


ME  CHA  NIC  A  L   DRA  WING. 


radius  cut  off  A2  in  C.  From  C  with  the  same  radius  cut  A$ 
in  I).  From  D,  A^  in  E.  From  B,  A$  in  F.  Join  A  C,  CDr 
DE,  £F,  and  FB. 

FIG,  42.  To  CONSTRUCT  A  REGULAR  HEPTAGON,  THE: 
CIRCUMSCRIBING  CIRCLE  BEING  GIVEN. — Draw  a  radius  AB. 
With  B  as  center  and  BA  as  radius,  cut  the  circumference  in 
1,2;  it  will  be  bisected  by  the  radius  in  C.  C\  or  €2  is  equal 
to  the  side  of  the  required  heptagon. 


FIG.  42. 


FIG    43. 


FIG.  43.  To  CONSTRUCT  A  REGULAR  OCTAGON,  THE- 
CIRCUMSCRIBING  CIRCLE  BEING  GIVEN. — Draw  a  diameter 
AB.  'Bisect  the  arcs  AB  in  C  and  D.  Bisect  arcs  CA  and 
CB  in  i  and  2.  Draw  lines  from  I  and  2  through  the  center 
of  the  circle,  cutting  the  circumference  in  3  and  4.  Join  Ai, 
iCj  C2,  2By  B$r  etc. 

FIG.  44.  To  CONSTRUCT  A  PENTAGON,  THE  SIDE  AB 
BEING  GIVEN. — Produce  AB.  With  B  as  center  and  BA  as 
radius,  describe  arc  AD2.  With  center  A  and  same  radius, 
describe  an  arc  cutting  the  first  arc  in  D.  Bisect  AB  in  E~ 


GEOMETRICAL   DRAWING.  2g 

Draw  line  DE.  Bisect  arc  BD  in  F.  Draw  line  EF.  With 
center  C  and  radius  EF  cut  off  arc  Ci  and  i,  2  on  the  semi- 
circle. Draw  line  B2  ;  it  will  be  a  second  side  of  the  penta- 


FIG.  44. 

gon.  Bisect  it  and  draw  a  line  perpendicular  to  it  at  the 
bisection.  The  perpendiculars  from  the  sides  AB  and  B2 
will  cut  in  G.  With  G  as  center  and  radius  GA  describe  a 
circle  *  it  will  contain  the  pentagon. 


FIG.  45. 


MECHANICAL   DRAWING. 


FIG.  45.  To  CONSTRUCT  A  HEPTAGON  ON  A  GIVEN 
LINE  AB. — Extend  line  AB  to  C.  From  B  with  radius  AB 
describe  a  semicircle.  With  center  A  and  same  radius  de- 
scribe an  arc  cutting  the  semicircle  in  D.  Bisect  AB  in  E. 
Draw  line  DE.  With  C  as  center  and  DE  as  radius,  cut  off 
arc  I  on  the  semicircle.  Draw  line  B\  ;  it  is  a  second  side  of 
the  heptagon.  Bisect  it  and  obtain  the  center  of  the  circum- 
scribing circle  as  in  the  preceding  problem. 

FIG.  46.  To  INSCRIBE  AN  OCTAGON  IN  A  GIVEN 
SQUARE. — Draw  diagonals  AD,  CB  intersecting  at  O.  From 
A,  B,  C,  and  D  with  radius  equal  to  AO  describe  quadrants 
cutting  the  sides  of  the  square  in  i,  2,  3,  4,  5,  6,  7,  8.  Join 
these  points  and  the  octagon  will  be  inscribed. 

C  6  R  7)  5 


FIG.  46. 

FIG.  47.  To  CONSTRUCT  A  REGULAR  OCTAGON  ON  A 
GIVEN  LINE  AB. — Extend  line  AB  in  both  directions.  Erect 
perpendiculars  at  A  and  B.  With  centers^  and  B  and  radius 
AB  describe  the  semicircle  CEB  and  AF2.  Bisect  the  quad- 
rants CE  and  DF  in  I  and  2,  then  Ai  and  B2  will  be  two 
more  sides  of  the  octagon.  At  I  and  2  erect  perpendiculars 
I,  3  and  2,  4  equal  to  AB.  Draw  1-2  and  3-4.  Make  the 


GEOMETRICAL   DRAWING.  31 

perpendiculars  at  A  and  B  equal  to  1-2  or  3-4 — viz.,  A$  and 
B6.     Complete  the  octagon  by  drawing  3-5,  5-6,  and  6-4. 

FIG.  48.  To  DRAW  A  RIGHT  LINE  EQUAL  TO  HALF 
THE  CIRCUMFERENCE  OF  A  GIVEN  CIRCLE. — Draw  a  diam- 
eter AB.  Draw  line  AC  perpendicular  to  AB  and  equal  to 
three  times  the  radius  of  the  circle.  Draw  another  perpen- 
dicular at  B  to  AB.  With  center  B  and  radius  of  the  circle 
cut  off  arc  BD,  bisect  it  and  draw  a  line  from  the  center  of 
the  circle  through  the  bisection,  cutting  line  B  in  E.  Join 
EC.  Line  EC  will  be  equal  to  half  the  circumference  of 
circle  A. 


FIG.  49.  To  FIND  A  MEAN  PROPORTIONAL  TO  TWO 
GIVEN  RIGHT  LINES. — Extend  the  line  AB  to  E  making  BE 
equal  to  CD.  Bisect  AE  in  F*  From  F  with  radius  FA  de- 
scribe a  semicircle.  At  B  where  the  two  given  lines  are 
joined  erect  a  perpendicular  to  AE  cutting  the  semicircle  in 
G.  BG  will  be  a  mean  proportional  to  CD  and  AB. 

FIG.  50.  To  FIND  A  THIRD  PROPORTIONAL  (LESS)  TO 
TWO  GIVEN  RIGHT  LINES  AB  AND  CD. — Make  EF=.the 
given  line  AB.  Draw  EG  making  an  angle  with  EF=  DC. 
Join  FG.  From  E  with  EG  as  radius  cut  EF  in  H.  Draw 


MECHANICAL   DRA  WING. 


H  parallel  to  FG,  cutting  EG  in  /.     El  is  the  third  propor- 
tional (less)  to  the  two  given  lines. 


C  D  £  F 

FIG.  50.  FIG.  51. 

FIG.  51.  To  FIND  A  FOURTH  PROPORTIONAL  TO  THREE 
GIVEN  RIGHT:  LINES  AB,  CD,  AND  EF.— Make  67/=the 
given  line  AB.  Draw  GI  =  CD,  making  any  convenient 
angle  to  GH.  Join  HI.  From  G  lay  off  GH —  EF.  From 
K  draw  a  parallel  to  HI  cutting  GI  in  L.  GL  is  the  fourth 
proportional  required. 


FIG.  52.  FIG.  53- 

FIG.  52.  To  FIND  THE  CENTER  OF  A  GIVEN  ARC  ABC. 
—Draw  the  chords  AB  and  CD    and   bisect  them.      Extend 
the  bisection  lines  to  intersect  in  D  the  center  required. 


GEOMETRICAL   DRAWING. 


33 


FIG.  5 3. "To-  DRAW  A  LINE  TANGENT  TO  AN  ARC  OF  A 
CIRCLE. -^(  i  st.)  'When  the  center  is  not  accessible.  Let  B 
be  the  point  through  which  the  tangent  is  to  be  drawn. 
From  B  lay  off  equal  distances  as  BE,  BF.  Join  EF  and 
through  B  draw  ABC  parallel  to  EF.  (2d.)  When  the  cen- 
ter D  is  given.  Draw  BD  and  through  B  draw  ABC  perpen- 
dicular to  BD-  '  ABC  is  tangent  to  ^heeifcte  at  the  point  B.~ 

FIG.  54.  To  DRAW  TANGENTS  TO  THE  CIRCLE  C  FROM 

THE  PoiNT^  WITHOUT  IT. — Draw  \AC  and   bisect  it  in  E. 

\      *>. 
From  E  with  radius  EC  describe  an  arcf  cutting  circle  C  in  B 

and  D.     Join  CB,  CD.      Draw  AB  and  AD  tangent   to   the 
•circle  C. 


FIG.  54.  FIG.  55. 

FIG.  55.  To  DRAW  A  TANGENT  BETWEEN  TWO  CIR- 
CLES.— Join  the  centers  A  and  B.  Draw  any  radial  line 
from  A  as  A2  and  make  1-2  =  the  radius  of  circle  B.  From 
A  with  radius  A-2  describe  a  circle  CiD.  From  center  B 


34 


MECHANICAL   DRAWING. 


draw  tangents  BC  and  BD  to  circle  C2D  at  the  points  C  and 
D  by  preceding  problem.  Join  AC  and  AD  and  through 
the  points  E  and  F  draw  parallels  ^6"  and  EH to  j5Z>  and  j#<7. 
/*"£  and  EH  are  the  tangents  required. 

FIG.  56.  To  DRAW  TANGENTS  TO  TWO  GIVEN  CIR- 
CLES A  AND  B. — Join  A  and  B.  From  A  with  a  radius 
equal  to  the  difference  of  the  radii  of  the  given  circles  de- 


FIG.   56. 


FIG.  57- 


scribe  a  circle  GF.  From  B  draw  the  tangents  BF  and  BG, 
by  Prob.  37.  Draw  AF  and  ^£  extended  to  E  and  //. 
Through  E  and  //  draw  £C  and  HD  parallel  to  ^  and  BG 
respectively.  EC  and  DH  are  the  tangents  required. 

FIG.  57.  To  DRAW  AN  ARC  OF  A  CIRCLE  OF  GIVEN 
RADIUS  TANGENT  TO  TWO  STRAIGHT  LINES. — AB  and  AC 
are  the  two  straight  lines,  and  r  the  given  radius.  At  a  dis- 
tance =  r  draw  parallels  1-2  and  3-4  to  AC  and  y2,#,  inter- 


Or 

GEOMETRICAL  DRAWING. 


35 


secting  at  F.  From  F  draw  perpendiculars  FD  and  FE. 
With  /*  as  center  and  FD  or  .faE  as  radius  describe  the  re- 
quired arc,  which  will  be  tangent  to  the  two  straight  lines  at 
the  points  D  and  E. 

FIG.  58.  To  DRAW  AN  ARC  OF  A  CIRCLE  TANGENT 
TO  TWO  STRAIGHT  LINES  BC  and  CD  WHEN  THE  MID- 
POSITION  G  IS  GIVEN. — Draw  CA  the  bisection  of  the  angle 
BCD  and  EF  at  right  angles  to  it  through  the  given  point  G. 
Next  bisect  either  of  the  angles  FEB  or  EFD.  The  bisection 
line  will  intersect  the  central  line  CA  at  A,  which  will  be  the 
center  of  the  arc.  From  A  draw  perpendiculars  Ai  and  A2, 
and  with  either  as  a  radius  and  A  as  center  describe  an  arc 
which  will  be  tangent  to  the  lines  BC  and  CD  at  the  points  I 
and  2. 

'»  A 


FIG.  58. 

FIG.  59.  To  INSCRIBE  A  CIRCLE  WITHIN  A  TRIANGLE 
ABC. — Bisect  the  angles  A  and  B.  The  bisectors  will  meet 
in  D.  Draw  Di  perpendicular  to  AB.  Then  with  center  D 
and  radius  =  D\  describe  a  circle  which  will  be  tangent  to 
the  given  triangle  at  the  points  I,  2,  3. 

FIG.  60.    To  DRAW  AN  ARC  OF   A  CIRCLE  OF  GIVEN 
RADIUS  R  TANGENT  TO  TWO  GIVEN  CIRCLES  A  AND  B.— 
From  A  and  B  draw  any  radial  lines  as  ^3,  B^..     Outside 
the  circumference  of  each  circle  cut  off  distances  1-3  and  2-4 


30  MECHANICAL   DRAWING. 

each  —  the  given  radius  R.  Then  with  center  A  and  radius 
A-$,  and  center  B  and  radius  £-4  describe  arcs  intersecting  at 
C.  Draw  CA,CB  cutting  the  circles  at  5  and  6.  With  centre 
C  and  radius  C^  or  C6  describe  an  arc  which  will  be  tangent 
at  points  5  and  6. 


FIG.  60. 

FIG.  61.  To    DRAW  AN  ARC  OF  A  CIRCLE  OF   GIVEN 
RADIUS  R  TANGENT  TO  TWO   GIVEN   CIRCLES  A   AND  B 


WHEN  THE  ARC  INCLUDES  THE  CIRCLES. — Through  A  and  B 
draw  convenient  diameters  and  extend  them  indefinitely.    On 


GEOMETRICAL  DRAWING. 


37 


these  measure  off  the  distances  1-2  and  3-4,  each  equal  in 
length  to  the  given  radius  R.  Then  with  center  A  and  radius 
A2,  center  B  and  radius  ^4,  describe  arcs  cutting  at  C.  From 
C  draw  C$  and  C6  through  B  and  A.  With  center  C  and  ra- 
dius C6  or  C$  describe  the  arc  6,  5,  which  will  be  tangent  to 
the  circles  at  the  points  6  and  5. 

FIG.   62.   To   DRAW  AN  ARC  OF  A  CIRCLE  .OF  GIVEN 
RADIUS  R  TANGENT  TO   Two   GIVEN   CIRCLES  A    AND  B 

WHEN  THE  ARC  INCLUDES  ONE  CIRCLE  AND  EXCLUDES  THE 
OTHER. — Through  A  draw  any  diameter  and  make  1-2  =  R. 


FIG.  62. 

From  B  draw  any  radius  and  extend  it,  making  3—4  =  R.  With 
center  A  and  radius  A2  and  center  B  and  radius  B^  describe 
arcs  cutting  at  C.  With  C  as  center  and  radius  ==  C$  or  C6 
describe  the  arc  5,  6. 

FIG.  63.  DRAW  AN  ARC  OF  A  CIRCLE  OF  GIVEN  RA- 
DIUS R  TANGENT  TO  A  STRAIGHT  LlNE  AB  AND  A  CIRCLE 
CD. — From  E,  the  center  of  the  given  circle,  draw  an  arc  of  a 


38  MECHANICAL   DRAWING. 

circle  i,  2  concentric  with  CD  at  a  distance  R  from  it,  and 
also  a  straight  line  3,  4  parallel  to  AB  at  the  same  distance  R 
from  AB.  Draw  £<9  intersecting  CD  at  5.  Draw  the  perpen- 
dicular (96.  With  center  O  and  radius  (96  or  (95  describe  the 
required  arc. 


FIG.  63. 

FIG.  64.   To   DESCRIBE  AN    ELLIPSE   APPROXIMATELY 
BY    MEANS    OF  THREE  RADII      (F.  R.   Honey's  method).— 


FIG.   64. 

Draw  straight  lines  RH  and  //(2,  making  any  convenient  angle 
at  //.      With  center  H  and  radii  equal  to  the  semi-minor  and- 


GEOMETRICAL   DRAWING. 


39 


semi-major  axes  respectively,  describe  arcs  LM  and  NO.  Join 
LO  and  draw  MK  and  NP  parallel  to  LO.  Lay  off  Li  =  i 
of  LN.  Join  (9i  and  draw  M2  and  7V3  parallel  to  Oi.  Take 
//3  for  the  longest  radius  (=  T),  H2  for  the  shortest  radius 
{=  £"),  and  one-half  the  sum  of  the  semi-axes  for  the  third 
radius  (=  5),  and  use  these  radii  to  describe  the  ellipse  as 
follows:  Let  AB  and  CD  be  the  major  and  minor  axes.  Lay 
off  A4  =  E  and  AS  =  5.  Then  lay  off  CG  =  T  and  C6  =  5. 
With  G  as  center  and  G6  as  radius  draw  the  arc  6,  g.  With 
center  4  and  radius  4,  5,  draw  arc  5,  g,  intersecting  6,  g  at  g. 
Draw  the  line  Gg  and  produce  it  making  £8  =  T.  Draw  g, 
4  and  extend  it  to  7  making  £-,  7=5.  With  center  £  and 
radius  GC(=T)  draw  the  arc  C&.  With  center  £•  and  radius 
g,  8  (=5)  draw  the  arc  8,  7.  With  center  4  and  radius  4,  7 
(=  E)  draw  arc  7^4.  The  remaining  quadrants  can  be  drawn 
in  the  same  way. 

FIG.  65.  To  DRAW  AX  ELLIPSE  HAVING  GIVEN  THE 
AXES  AB  AND  CD. — Draw  AB  and  CD  at  right  angles  to  and 
bisecting  each  other  at  E.  With  center  C  and  radius  EA  cut 
AB  in  F  and  F  the  foci.  Divide  EF  or  EF'  into  a  number  of 
parts  as  shown  at  i.,  2,  3,  4,  etc.  Then  with  F  and  F'  as  cen- 


FIG.  65. 


FIG.  66. 


FIG.  67.. 


ters  and  ^4i  and  Bi,  and  ^2  and  ^2,  etc.,  as  radii  describe  arcs 
intersecting  in  R,  5,  etc.,  until  a  sufficient  number  of  points 


4O  MECHANICAL   DRAWING. 

are  found  to  draw  the  elliptic  curve  accurately  throughout. 
(No.  5  of  the  "Sibley  College  Set"  of  irregular  curves  is 
very  useful  in  drawing  this  curve.)  To  draw  a  tangent  to 
the  ellipse  at  the  point  G :  Extend  FG  and  draw  the  bisector 
of  the  angle  HGF '.  KG  is  the  tangent  required. 

FIG.  66.  ANOTHER  METHOD. — Let  AB  and  AC  be  the 
semi  axes.  With  A  as  center  and  radii  AB  and  AC  describe 
circles.  Draw  any  radii  as  A^  and  ^4,  etc.  Make  3  i,  42, 
etc.,  perpendicular  to  AB,  and  D2,  E*>,  etc.,  parallel  to  AB. 
Then  i,  2,  5,  etc.,  are  points  on  the  curve. 

FIG.  67.  ANOTHER  METHOD. — Place  the  diameters  as 
before,  and  construct  the  rectangle  CDEF.  Divide  AB  and 
DB  and  BF  into  the  same  number  of  equal  parts  as  i,  2,  3  and 
B.  Draw  from  C  through  points  i-,  2,  3  on  AB  and  BD 
lines  to  meet  others  drawn  from  E  through  points  i,  2,  3  on 
AB  and  FB  intersecting  in  points  GHK.  GHK  are  points  on 
the  curve. 

FIG.  68.  ANOTHER  METHOD. — Place  the  diameters  AB 
and   CD  as  shown  in  Drawing  No.  i.     Draw  any  convenient 
"i 


FIG.  68. 


angle  RHQ,  Drawing  No.  2.     With  center  //and  radii  equal 
to  the  semi-minor  and  semi-major  axes  describe  arcs  LM  and 


GEOMETRICAL   DRAWING.  4! 

NO.  Join  LO  and  draw  MK  and  NP  parallel  to  LO.  Then 
from  C  and  D  with  a  distance  =  HP  lay  off  the  points  I  i'  on 
the  minor  axis  and  from  A  and  B  with  a  distance  =  HK  lay 
off  the  points  2  2'  on  the  major  axis.  With  centers  I,  i',  2  and 
2'  and  radii  i-D  and  2-^,  respectively,  draw  arcs  of  circles, 
On  a  piece  of  transparent  celluloid  T  lay  off  from  the  point  6> 
GF  and  GE  =  the  semi-minor  and  semi-major  axes  respec- 
tively. Place  the  point  Fon  jthe  major  axis  and  the  point  E  on 
the  minor  axis.  If  the  strip  of  celluloid  is  now  moved  over 
the  figure,  so  that  the  point  E  is  always  .in  contact  with  the 
semi-minor  axis  and  the  point  .F  with  the  semi  major  axis,  the 
necessary  number  of  points  may  be  marked  through  a  smalL 
hole  in  the  celluloid  at  G  with  a  sharp  conical-pointed  pencil, 
and  thus  complete  the  curve  of  the  ellipse  between  the  arcs  of 
circles. 

FIG.  69.  To  CONSTRUCT  A  PARABOLA,  THE  BASE  CD 
AND  THE  ABSCISSA  AB  BEING  GIVEN.  —  Draw  EF  through  A 
parallel  to  CD  and  CE  and  DF  parallel  to  AB.  Divide  AEr 
AF,  EC,  and  FD  into  the  same  number  of  equal  parts. 
Through  the  points  I,  2,  3  on  AF  and  AE  draw  lines  parallel 
to  AB,  and  through  A  draw  lines  to  the  points  I,  2,  3  on  FD 
and  EC  intersecting  the  parallel  lines  in  points  4,  5,  6,  etc.,  of 
the  curve. 

FIG.  70.  GIVEN  THE  DIRECTRIX  BD  AND  THE  Focus  C 
TO  DRAW  A  PARABOLA  AND  A  TANGENT  TO  IT  AT  THE  POINT 
3. — The  parabola  is  a  curve  such  that  every  point  in  the  curve 
is  equally  distant  from  the  directrix  j#Z>  and  the  focus  C.  The 
vertix  E  is  equally  distant  from  the  directrix  and  the  focus, 
i.e.  CE  is  =  EB.  Any  line  parallel  to  the  axis  is  a  diameter. 
A  straight  line  drawn  across  the  figure  at  right  angles  to  the 


MECHANICAL   DRA  WING. 


axis  is  a  double  ordinate,  and  either  half  of  it  is  an  ordinate. 
The  distance  from  C  to  any  point  upon  the  curve,  as  2  is 
always  equal  to  the  horizontal  distance  from  that  point  to  the 
directrix.  Thus  C\  =  I,  i' ',  €2  to  2,  2',  etc.  Through  C 
draw  ACF  at  right  angles  to  J5D,  ACF  is  the  axis  of  the 


3  F 


D 


FIG.  70. 


curve.  Draw  parallels  to  BD  through  any  points  in  AB,  and 
with  center  C  and  radii  equal  to  the  horizontal  distances  of 
these  parallels  from  BD  describe  arcs  cutting  in  the  points  I, 
2,  3,  4,  etc.  These  are  points  in  the  curve.  The  tangent  to 
the  curve  at  the  point  3  may  be  drawn  as  follows :  Produce 
AB  to  F.  Make  EF  =  the  horizontal  distance  of  ordinate  33 
from  E.  Draw  the  tangent  through  3^. 

FIG.  71.  To  DRAW  AN  HYPERBOLA,  HAVING  GIVEN 
THE  DIAMETER  AB,  THE  ABSCISSA  BD,  AND  DOUBLE  ORDI- 
NATE EF. — Make  F^  parallel  and  equal  to  BD.  Divide  DF 
and  /4  into  the  same  number  of  equal  parts.  From  B  draw 
lines  to  the  points  in  ^F  and  from  A  draw  lines  to  the  points 
in  DF.  Draw  the  curve  through  the  points  where  the  lines 
correspondingly  numbered  intersect  each  other. 


GEOMETRICAL   DRAWING. 


43 


FIG.  72.  To  CONSTRUCT  AN  OVAL  THE  WIDTH  AB 
BEING  GIVEN.— Bisect  AB  by  the  line  CD  in  the  point  E, 
and  with  E  as  center  and  radius  EA  draw  a  circle  cutting  CD  in 


FIG.  71.  FIG.  72. 

.F.    From  ^  and  B  draw  lines  through  /^    From  A  and  ^  with 
radius  equal  to  AB  draw  arcs  cutting  the  last  two  lines  in  G 


and  H.    From  F  with  radius  /^  describe  the  arc  677  to  meet 
the  arcs  AG  and  BH,  which  will  complete  the  oval. 

FIG.  73.  GIVEN  AN  ELLIPSE  TO  FIND  THE  AXES  AND 
Foci.  —  Draw  two  parallel  chords  AB  and  CD.  Bisect  each 
of  these  in  E  and  F.  Draw  EF  touching  the  ellipse  in  i  and 
2.  This  line  divides  the  ellipse  obliquely  into  equal  parts. 
Bisect  I,  2  in  6",  which  will  be  the  center  of  the  ellipse.  From 
G  with  any  radius  draw  a  circle  cutting  the  ellipse  in  HIJK. 
Join  these  four  points  and  a  rectangle  will  be  formed  in  the 
ellipse.  Lines  LM  and  NO,  bisecting  the  sides  of  the 
rectangle,  will  be  the  diameters  or  axes  of  the  ellipse.  With 
N  or  O  as  centers  and  radius  =  GL  the  semi-major  axis,  de- 
scribe arcs  cutting  the  major  axis  in  P  and  Q  the  foci. 

FIG.  74.   To  CONSTRUCT  A   SIPRAL  OF  ONE  REVOLU- 
TION. —  Describe  a  circle  using  the  widest  limit  of  the  spiral  as 


44 


MECHANICAL   DRA  WING. 


a  radius.  Divide  the  circle  into  any  number  of  equal  parts  as 
A,  B,  C,  etc.  Divide  the  radius  into  the  same  number  of  equal 
parts  as  I  to  12.  From  the  center  with  radius  12,  I  describe 
an  arc  cutting  the  radial  line  B  in  \' .  From  the  center  con- 
tinue to  draw  arcs  from  points  2,  3,  4,  etc.,  cutting  the  corre- 
sponding radii  C,  D,  E,  etc.  in  the  points  2' ',  3',  4',  etc.  From 
12  trace  the  Archimedes  Spiral  of  one  revolution. 


FIG.  75.  To  DESCRIBE  A  SPIRAL  OF  ANY  NUMBER  OF 
REVOLUTIONS,  E.G.,  2. — Divide  the  circle  into  any  num- 
ber of  equal  parts  as  A,  B,  C,  etc.,  and  draw  radii.  Divide 
the  radius  A 12  into  a  number  of  equal  parts  corresponding 
with  the  required  number  of  revolutions  and  divide  these 
into  the  same  number  of  equal  parts  as  there  are  radii,  viz., 
i  to  12.  It  will  be  evident  that  the  figure  consists  of  two 
separate  spirals,  one  from  the  center  of  the  circle  to  12,  and 
one  from  12  to  A.  Commence  as  in  the  last  problem,  draw- 
ing arcs  from  I.  2,  3,  etc.,  to  the  correspondingly  numbered 
radii,  thus  obtaining  the  points  marked  i',  2',  3',  etc.  The 
first  revolution  completed,  proceed  in  the  same  manner  to 
find  the  points  i",  2",  3" ',  etc.  Through  these  points  trace 
the  spiral  of  two  revolutions. 


GEOMETRICAL   DRAWING. 


45 


FIG.  76.  To  CONSTRUCT  THE  INVOLUTE  OF  THE  CIR- 
CLE O. — Divide  the  circle  into  any  number  of  equal  parts 
and  draw  radii.  Draw  tangents  at  right  angles  to  these  radii. 
On  the  tangent  to  radius  I  lay  off  a  distance  equal  to  one 
of  the  parts  into  which  the  circle  is  divided,  and  on  each  of 


the  tangents  set  off  the  number  of  parts  corresponding  to  the 
number  of  the  radii.  Tangent  12  will  then  be  the  circumfer- 
ence of  the  circle  unrolled,  and  the  curve  drawn  through  the 
extremities  of  the  other  tangents  will  be  the  involute. 

FIG.  77.  To  DESCRIBE  AN  IONIC  VOLUTE. — Divide  the 
given  height  into  seven  equal  parts,  and  through  the  point  3 
the  upper  extremity  of  the  third  division  draw  3,  3  perpen- 
dicular to  AB.  From  any  convenient  point  on  33  as  a  cen- 
ter, with  radius  equal  to  one-half  of  one  of  the  divisions  on 
AB,  describe  the  eye  of  the  volute  NPNM,  shown  enlarged 
at  Drawing  No.  2.  NN  corresponds  to  line  3,  3,  Drawing 
No.  i.  Make  PM  perpendicular  to  NN  and  inscribe  the 
square  NPNM^  bisect  its  sides  and  draw  the  square  n,  12, 


46 


ME  CHA  A7fCA  L   DRA  WING. 


13,  14.  Draw  the  diagonals  n,  13  and  12,  14  and  divide 
them  as  shown  in  Drawing  No.  2.  At  the  intersections  of 
the  horizontal  with  the  perpendicular  full  lines  locate  the 
points  i,  2,  3,  4,  etc.,  which  will  be  the  centers  of  the  quad- 
rants of  the  outer  curve.  The  centers  for  the  inner  curve 
will  be  found  at  the  intersections  of  the  horizontal  and  per- 


FIG.   77- 

pendicular  broken  lines,  drawn  through  the  divisions  on  the 
diagonals.  Then  with  center  I  and  radius  I  Pel  raw  arc  PN, 
.and  with  center  2  and  radius  2^Vdraw  arc  NM,  with  center  3 
and  radius  ^M  draw  arc  ML,  etc.  The  inner  curve  is  drawn 
in  a  similar  way,  by  using  the  points  on  the  diagonals  indi- 
cated by  the  broken  lines  as  centers. 

FIG.  78.  To  DE-CRIBE  THE  CYCLOID. — AB  is  the  di- 
rector, CB  the  generating  circle,  X  a  piece  of  thin  transparent 
celluloid,  with  one  side  dull  on  which  to  draw  the  circle  C. 
At  any  point  on  the  circle  C  puncture  a  small  hole  with  a 
sharp  needle,  and  place  the  point  C  tangent  to  the  director 
AB  at  the  point  from  which  the  curve  is  to  be  drawn.  Hold 
.the  ce.lluloid  at  this  point  with  a  needle,  and  rotate  it  until 


GEOMETRICAL   DRAWING. 


47 


the  arc  of  the  circle  C  intersects  the  director  AB.  Through 
the  point  of  intersection  stick  another  needle  and  rotate  X 
until  the  circle  is  again  tangent  to  AB,  and  through  the  punc- 
ture at  C  with  a  4-H  pencil,  sharpened  to  a  fine  conical  point, 
mark  the  first  point  on  the  curve.  So  proceed  until  sufficient 
points  have  been  found  to  complete  the  curve. 

(NOTE. — The  thin  celluloid  was  first  used  as  a  drawing; 
instrument  by  Professor  H.  D.  Williams,  of  Sibley  College, 
Cornell  University.) 

FIG.  79.  To  FIND  THE  LENGTH  OF  A  GIVEN  ARC  OF  A 
CIRCLE  APPROXIMATELY.— Let  BC  be  the  given  arc.  Draw 
its  chord  and  produce  it  to  A,  making  BA  equal  half  the 


FIG.  78. 


chord.  With  center^  and  radius  AC  describe  arc  CD  cut- 
ting the  tangent  line  BD  at  D,  and  making  it  equal  to  the 
arc  BC. 

FIG.  80.  To  DESCRIBE  THE  CYCLOID  BY  THE  OLD 
METHOD  — Divide  the  director  and  the  generating  circle  into 
the  same  number  of  equal  parts.  Through  the  center  a  draw 
ag  parallel  to  AB  for  the  line  of  centers,  and  divide  it  as  AB 
in  the  points  £,  c,  d,  e,f,  and  g.  With  centers/,  e,  d,  etc.,  de- 
scribe arcs  tangent  to  AB,  and  through  the  points  of  division 
on  the  generating  circle  I,  2,  3,  etc.,  draw  lines  parallel  to 


48 


MECHANICAL   DRA  WING. 


AB  cutting  the  arcs  in  the  points  i',  2',  3',  etc.     These  will  be 
points  in  the  curve. 

An  approximate  curve  may  be  drawn  by  arcs  of   circles. 
Thus,  taking/7  as  center  and  f ' g'  as  radius,  draw  arc  g'if. 


FIG.  80. 

Produce  \'f  and  2 ' e'  until  they  meet  at  the  center  of  the 
second  arc  2'f,  etc. 

FIG.  81.  To  DESCRIBE  THE  EPICYCLOID  AND  THE 
HYPOCYCLOID. — Divide  the  generating  circle  into  any  num- 
ber of  equal  parts,  I,  2,  3,  etc.,  and  set  off  these  lengths  from 
C  on  the  directing  circle  CB  as  c' ,  d' ,  c' ,  etc.  From  A  the  cen- 
ter of  the  directing  circle  draw  lines  through  /,  ci ',  c',  etc.,  cut- 
ting the  circles  of  centers  in  e,  d,  c,  etc.  From  each  of  these 
points  as  centers  describe  arcs  tangent  to  the  directing  circle. 
From  center  A  draw  arcs  through  the  points  of  division  on 
the  generating  circle,  cutting  the  arcs  of  the  generating  circles 
in  their  several  positions  at  the  points  i',  2',  3',  etc.  These 
will  be  points  in  the  curve. 

Fie;.  82.  ANOTHER  METHOD. — Dra\v  the  generating 
circle  on  the  celluloid  and  roll  it  on  the  outside  of  the  gener- 
ating circle  BC  for  the  Epicycloid,  and  on  the  inside  for  the 


GEOMETRICAL   DRAWING. 


49 


Hypocycloid,  marking  the  points  in  the  curve  1,2,  3,  etc.,  in 
similar  manner  to  that  described  for  the  Cycloid. 


FIG.  82. 


FIG.  81. 


FIG.  83. 


FIG.  83.  To  DRAW  THE  CISSOID. — Draw  any  line  AB 
and  BC  perpendicular  to  it.  On  BC  describe  a  circle.  From 
the  extremity  C  of  the  diameter  draw  any  number  of  lines, 
at  any  distance  apart,  passing  through  the  circle  and  meeting* 
the  line  AB  in  i', '2',  3',  etc.  Take  the  length  from  A  to  9 
and  set  it  off  from  C  on  the  same  line  to  9" '.  Take  the  dis- 
tance from  8'  to  8  and  set  it  off  from  C  on  the  same  line  to 
8",  etc.,  for  the  other  divisions,  and  through  9",  8''',  /',  6", 
•etc.,  draw  the  curve. 


MECHANICAL   DRA  WING. 


FIG.  84.  To  DRAW  SCHIELE'S  ANTI-FRICTION  CURVE. 
—Let  AB  be  the  radius  of  the  shaft  and  B\,  2,  3,  4,  etc.,  its 
axis.  Set  off  the  radius  AB  on  the  straight  edge  of  a  piece 
of  stiff  paper  or  thin  celluloid  and  placing  the  point  B  on  the 
division  I  of  the  axis,  draw  through  point  A  the  line  A\. 
Then  lower  the  straight  edge  until  the  point  B  coincides  with 
2  and  the  point  A  just  touches  the  last  line  drawn,  and  draw 
a2,  and  so  proceed  to  find  the  points  a,  b,  c,  etc.  Through 
these  points  draw  the  curve. 


4    5 


FIG.  84. 


FIG.  85. 


FIG.  85.    To  DESCRIBE    AN   INTERIOR   EPICYCLOID. 

Let  the  large  circle  X  be  the  generator  and  the  small  circle 
Y  the  director.  Divide  circle  Y  into  any  number  of  equal 
parts,  as  B,  H,  /,  J,  etc.  Draw  radial  lines  and  make  HC, 
ID,  JE,  KFy  etc.,  each  equal  to  the  radius  of  the  generator 
X.  With  centers  C,  D,  E,  etc.,  describe  arcs  tangent  at 
H,  I,J,  etc.  Make  Hi  equal  to  one  of  the  divisions  of  the  di- 
rector as  BH.  Make  /2  equal  to  two  divisions,  J$,  three  divi- 
sions, etc.,  and  draw  the  curve  through  the  points  i,  2,  3,  4, 


GEOMETRICAL   DRAWING.  51 

etc.     This  curve  may  also  be  described  with  a  piece  of  cellu- 
loid in  a  similar  way  to  that  explained  for  the  cycloid. 

It  may  not  be  out  of  place  here  to  describe  a  few  of  the 

MOULDINGS    USED    IN   ARCHITECTURAL   WORK, 

since  they  are  often  found  applied  to  mechanical  constructions. 
FIG.  86.  To  DESCRIBE  THE  "  SCOTIA."  —  i,  i  is  the  top 
line  and  4,  4  the  bottom  line.  From  i  drop  a  perpendicular 
I,  4;  divide  this  into  three  equal  parts,  as  I,  2,  and  3. 
Through  the  point  2  draw  ab  parallel  to  I,  i.  With  center  2 
and  radius  2,  i  describe  the  semicircle  a\b,  and  with  center  b 
and  radius  ba  describe  the  arc  #5  tangent  to  4,  4  at  5,  draw 
the  fillets  i,  i  and  4,  4. 


~~i 


3 

\ 

FIG.  86.  FIG.  87. 

FIG.  87.  To  DESCRIBE  THE  «CYMA  RECTA." — Join  ir 
3  and  divide  it  into  five  equal  parts,  bisect  i,  2  and  2,  3,  and 
with  radius  equal  to  I,  2  and  2,  3  respectively  describe  arcs 
i,  2  and  2,3.  Draw  the  fillets  I,  I  and  3,  3  and  complete  the 
moulding. 

FIG.  88.  To  DESCRIBE  THE  "CAVETTO"  OR  "  HOL- 
LOW."— Divide  the  perpendicular  i,  2  into  three  equal  parts 
and  make  2,  3  equal  to  two  of  these.  From  centers  I  and  3 
with  a  radius  somewhat  greater  than  the  half  of  I,  3,  describe 
arcs  intersecting  at  the  center  of  the  arc  1,3, 


MECHANICAL   DRAWING. 


FIG.  89.  To  DESCRIBE  THE  "  ECHINUS,'  "  QUARTER 
ROUND,"  OR  "OvOLO."-— Draw  I,  2  perpendicular  to  2,  3, 
and  divide  it  into  three  equal  parts.  Make  2,  3  equal  to 
two  of  these  parts.  From  the  points  2  and  3  with  a  radius 
greater  than  half  1,3,  describe  arcs  cutting  in  the  center  of 
the  required  curve. 


FIG.  90.  To  DESCRIBE  THE  "  APOPHYGEE." — Divide 
3,  4  into  four  equal  parts  and  lay  off  five  of  these  parts  from 
3  to  2.  From  points  2  and  4  as  centers  and  radius  equal  to 
2,3,  describe  arcs  intersecting  in  the  center  of  the  curve. 


FIG.  91. 

FIG.  91.  To  DESCRIBE  THE  "  CYMA  REVERSA." — Make 
4,  3  =  4,  i.  Join  i,  3  and  bisect  it  in  the  point  2.  From  the 
points  i,  2  and  3  as  centers  and  radii  equal  to  about  two-thirds 
of  i ,  2  draw  arcs  intersecting  at  5  and  6.  Points  5  and  6 
are  the  centers  of  the  reverse  curves. 

FIG.  92.  To  DESCRIBE  THE  "  TORUS." — Let  i,  2  be  the 
breadth.  Drop  the  perpendicular  i,  2,  and  bisect  it  in  the 


GEOMETRICAL   DRAWING. 


53 


point  3.    With  3  as  center  and  radius  3,  I,  describe  the  semi- 
circle.     Draw  the  fillets. 


FIG.  92. 


FIG.  93. 


FIG.  93.  AN  ARCHED  WINDOW  OPENING. — The  curves 
are  all  arcs  of  circles,  drawn  from  the  three  points  of  the  equi- 
lateral triangle,  as  shown  in  the  figure. 

FIG.  94.  To  DESCRIBE  THE  "TREFOIL." — The  equi- 
lateral triangle  is  drawn  first,  and  the  angle  1,2,3  bisected  by 
the  line  2,  4,  which  also  cuts  the  perpendicnlar  line  I,  6  in  the 
point  6.  The  center  of  the  surrounding  circles  I,  2  and  3  are 
the  centers  of  the  trefoil  curves. 

FIG.  95.  To  DESCRIBE  THE  "  QUATRE  FOIL." — Draw 
the  square  I,  2,  3,  4  in  the  position  shown  in  the  figure.  The 
center  of  the  surrounding  circles,  point  5,  is  at  the  intersection 
of  the  diagonals  of  the  square.  Points  I,  2,  3,  4  of  the  square 
are  the  centers  of  the  small  arcs. 

FIG.  96.  To  DESCRIBE  THE  "CINQUEFOIL  ORNA- 
MENT." The  curves  of  the  cinquefoil  are  described  from  the 
corners  of  a  pentagon  1,2,3,  4>  5-  Bisect  4,  5  in  6  and  draw 
2,  6,  cutting  the  perpendicular  in  the  point  7,  the  center  of 
the  large  circles. 

FIG.  97.  To  DRAW  A  BALUSTER. — Begin  by  drawing 
the  center  line,  and  lay  off  the  extreme  perpendicular  height, 


54 


MECHANICAL  DRAWING. 


the   intermediate,    perpendicular,    and   horizontal  dimensions, 
and  finally  the  curves  as  shown  in  the  figure. 


FIG.  94. 


FIG.  95. 


FIG.  96. 


FIG.  97. 


DRAWING    TO    SCALE. 

When  we  speak  of  a  drawing  as  having  been  made  to  scale, 
we  mean  that  every  part  of  it  has  been  drawn  proportionately 
and  accurately,  either/////  size,  reduced  or  enlarged. 

Very  small  and  complicated  details  of  machinery  are  usu- 
ally drawn  enlarged ;  larger  details  and  small  machines  may 
be  made  full  size,  while  larger  machines  and  large  details  are 
shown  reduced. 

When  a  drawing  of  a  machine  is  made  to  a  reduced  or  en- 
larged scale  the  figures  placed  upon  it  should  always  give  the 
full-size  dimensions,  i.e.,  the  sizes  the  machine  should  meas- 
ure when  finished. 


GEOMETRICAL  DRAWING. 


55 


FIG.  98.  To  CONSTRUCT  A  SCALE  OF  THIRD  SIZE  OR 
4"=  I  FOOT. — Draw  upon  a  piece  of  tough  white  drawing- 
paper  two  parallel  lines  about ' \"  apart  and  about  14"  long  as 
shown  by  a,  Fig.  98.  From  A  lay  off  distances  equal  to  4" 
and  divide  the  first  space  AB  into  12  equal  parts  or  inches  by 
Prob.  12.  Divide  AE'm  the  same  way  into  as  many  parts  as 
it  may  be  desired  to  subdivide  the  inch  divisions  on 


n'  io'        8'    7' 


5-  4' 


mmm 


Scale  ?'-lfoot. 


kiuiklM 

<7 

/Scale      1'*  Ifoot.      I 

k                    !                                            .  -        f  '  5/*                                                                             ^ 

FIG.  98. 

usually  8.  When  the  divisions  and  subdivisions  have  been 
carefully  and  lightly  drawn  in  pencil,  as  shown  by  a,  in  Fig. 
98,  then  the  lines  denoting  -J",  i",  £",  i",  and  3"  should  be 
carefully  inked  and  numbered  as  shown  by  (b).  By  a  further 
subdivision  a  scale  of  2"=  I  foot  may  easily  be  made  as  shown 
by  (<r)  in  Fig.  98. 


CHAPTER    III. 
CONVENTIONS. 

It  is  often  unnecessary  if  not  undesirable  to  represent  cer- 
tain things  as  they  would  actually  appear  in  a  drawing,  espe- 
cially when  much  time  and  labor  is  required  to  make  them 
orthographically  true. 

So  for  economic  reasons  draftsmen  have  agreed  upon  con- 
ventional methods  to  represent  many  things  that  would  other- 
wise entail  much  extra  labor  and  expense,  and  serve  no  par- 
ticular purpose. 

It  is  very  necessary,  however,  that  all  draftsmen  should 
know  how  to  draw  these  things  correctly,  for  occasions  will 
often  arise  when  such  knowledge  will  be  demanded ;  and  be- 
sides it  gives  one  a  feeling  of  greater  satisfaction  when  using 
conventional  methods  to  know  that  he  could  make  them  artis- 
tically true  if  it  was  deemed  necessary. 

STANDARD    CONVENTIONAL    SECTION    LINES. 

Conventional  section  lines  are  placed  on  drawings  to  distin- 
guish the  different  kinds  of  materials  used  when  such  drawings 
are  to  be  finished  in  pencil,  or  traced  for  blue  printing,  or  to 
be  used  for  a  reproduction  of  any  kind. 

Water-colors  are  nearly  always  used  for  finished  drawings 
and  sometimes  for  tracings  and  pencil  drawings. 

The  color  tints  can  be  applied  in  much  less  time  than  it 

56 


CONVENTIONS.  57 

takes  to  hatch-line  a  drawing.  So  that  the  color  method 
should  be  used  whenever  possible. 

FIG.  99. — This  figure  shows  a  collection  of  hatch-lined 
sections  that  is  now  the  almost  universal  practice  among 
draftsmen  in  this  and  other  countries,  and  may  be  considered 
standard. 

No.  I.  To  the  right  is  shown  a  section  of  a  wall  made  of 
rocks.  When  used  without  color,  as  in  tracing  for  printing, 
the  rocks  are  simply  shaded  with  India  ink  and  a  175  Gillott 
steel  pen.  For  a  colored  drawing  the  ground  work  is  made 
of  gamboge  or  burnt  umber.  To  the  left  is  the  conventional 
representation  of  water  for  tracings.  For  colored  drawings 
a  blended  wash  of  Prussian  blue  is  added. 

No.  2.  Convention  for  Marble. —  When  colored,  the 
whole  section  is  made  thoroughly  wet  and  each  stone  is  then 
streaked  with  Payne's  gray. 

No.  3.  Convention  for  Chestnut. —  When  colored,  a 
ground  wash  of  gamboge  with  a  little  crimson  lake  and  burnt 
umber  is  used.  The  colors  for  graining  should  be  mixed  in  a 
separate  dish,  burnt  umber  with  a  little  Payne's  gray  and 
crimson  lake  added  in  equal  quantities  and  made  dark  enough 
to  form  a  sufficient  contrast  to  the  ground  color. 

No.  4.  General  Convention  for  Wood. — When  colored  the 
ground  work  should  be  made  with  a  light  wash  of  burnt  sienna. 
The  graining  should  be  done  with  a  writing-pen  and  a  dark 
mixture  of  burnt  sienna  and  a  modicum  of  India  ink. 

No.  5.  Convention  for  Black  Walnut. — A  mixture  of 
Payne's  gray,  burnt  umber  and  crimson  lake  in  equal  quanti- 
ties is  used  for  the  ground  color.  The  same  mixture  is  used 
for  graining  when  made  dark  by  adding  more  burnt  umber. 


MECHANICAL  DRAWING. 


CONVENTIONS.  $9 

No.  6.  Convention  for  Hard  Pine. —  For  the  ground 
color  make  a  light  wash  of  crimson  lake,  burnt  umber,  and 
gamboge,  equal  parts.  For  graining  use  a  darker  mixture  of 
of  crimson  lake  and  burnt  umber. 

No.  7.  Convention  for  Building-stone.  —  The  ground 
color  is  a  light  wash  of  Payne's  gray  and  the  shade  lines  are 
added  mechanically  with  the  drawing-pen  or  free-hand  with 
the  writing-pen. 

No.  8.  Convention  for  EartJi. — Ground  color,  India  ink 
and  neutral  tint.  The  irregular  lines  to  be  added  with  a  writ- 
ing-pen and  India  ink. 

No.  9.  Section  Lining  for  Wrought  or  Malleable  Iron. — 
When  the  drawing  is  to  be  tinted,  the  color  used  is  Prussian 
blue. 

No.  10.  Cast  Iron. — These  section  lines  should  be  drawn 
equidistant,  not  very  far  apart  and  narrower  than  the  body 
lines  of  the  drawing.  The  tint  is  Payne's  gray. 

No.  1 1.  Steel. — This  section  is  used  for  all  kinds  of  steel. 
The  lines  should  be  of  the  same  width  as  those  used  for  cast- 
iron  and  the  spaces  between  the  double  and  single  lines  should 
be  uniform.  The  qolor  tint  is  Prussian  blue  with  enough  crim- 
son lake  added  to  make  a  warm  purple. 

No.  12.  Brass. — This  section  is  generally  used  for  all 
kinda  of  composition  brass,  such  as  gun-metal,  yellow  metal, 
bronze  metal,  Muntz  metal,  etc.  The  width  of  the  full  lines, 
dash  lines  and  spaces  should  all  be  uniform.  The  color  tint 
is  a  light  wash  of  gamboge. 

Nos.  13-20. — The  section  lines  and  color  tints  for  these 
numbers  are  so  plainly  given  in  the  figure  that  further  instruc- 
tion would  seem  to  be  superfluous. 


60  MECHANICAL   DRAWING. 

CONVENTIONAL    LINES. 

FlG.   100. — There  are  four  kinds: 

(i)  The  Hidden  Line. — This  line  should  be  made  of  short 
dashes  of  uniform  length  and  width,  both  depending  some- 
what on  the  size  of  the  drawing.  The  width  should  always 
be  slightly  less  than  the  body  lines  of  the  drawing,  and  the 

(v  ~~  *~~~ """ ~"       """"" ——————       ~"~~        ~~~      —  —  — —  —  —  —.•—. 


(2) 

(3) 

(4) 


FIG.  100. 

length  of  the  dash  should  never  exceed  •£•".  The  spaces 
between  the  dashes  should  all  be  uniform,  quite  small,  never 
exceeding  TV'-  This  line  is  always  inked  in  with  black  ink. 

(2)  The  Line  of  Jlfotion. — This    line    is    used   to   indicate 
point  paths.    The  dashes  should  be  made  shorter  than  those  of 
the  hidden  line,  just  a  trifle   longer  than    dots.      The   spaces 
should  of  course  be  short  .and  uniform. 

(3)  Center  Lines. — Most   drawings   of  machines  and  parts 
of  machines  are   symmetrical  about  their  center  lines.      When 
penciling  a  drawing  these  lines  may  be  drawn   continuous  and 
as  fine  as  possible,  but  on  drawings  for  reproductions  the  black- 
inked   line  should  be  a  long  narrow  dash  and  two  short  ones 
alternately.   When  colored  inks  are  used  the  center  line  should 
be  made  a  continuous  red  line  and  as  fine  as  it  is  possible  to 
make  it. 

(4)  Dimension    Lines  and  Line  of  Section. — These    lines- 
are   made  in  black  with  a  fine  long  dash  and  one  short  dash 
alternately.      In  color  they  should  be  continuous  blue   lines. 


CONVENTIONS. 


6l 


Colored  lines  should  be  used  wherever  feasible,  because  they 
are  so  quickly  drawn  and  when  made  fine  they  give  the  drawing 
a  much  neater  appearance  than  when  the  conventional  black 
lines  are  used.  Colored  lines  should  never  be  broken. 

CONVENTIONAL  BREAKS. 

FlG.  101. — Breaks  are  used  in  drawings  sometimes  to  indi- 
cate that  the  thing  is  actually  longer  than  it  is  drawn,  some- 


FIG.  TOI. 

times  to  show  the  shape  of  the  cross-section  and  the  kind  of 
material.      Those  given  in  Fig.  101  show  the  usual  practice. 

CROSS-SECTIONS. 

FIG.  1 02. — When  a  cross-section  of  a  pulley,  gear-wheel 
or  other  similar  object  is  required  and  the  cutting-plane  passes 
through  one  of  the  spokes  or  arms,  then  only  the  rim  and  hub 
should  be  sectioned,  as  shown  at  xx  No.  I  and  zz  No.  2,  and 
the  arm  or  spoke  simply  outlined.  Cross-sections  of  the  arms 
mny  be  made  as  shown  at  A  A  No.  2.  In  working  drawings  of 


MECHANICAL  DRAWING. 


gear-wheels  only  the  number  of  teeth  included  in  one  quadrant 
need  be  drawn  ;  the  balance  is  usually  shown  by  conventional 
lines,  e.g.,  the  pitch  line  the  same  as  a  center  line,  viz.,  a  long 


FIG.  102. 

dash  and  two  very  short  ones  alternately  or  a  fine  continuous 
red  line. 

The  addendum  line  (d)  and  the  root  or  bottom  line  (b)  the 
same  as  a  dimension  line,  viz.,  one  long  dash  and  one  short 
dash  alternately  or  a  fine  continuous  blue  line.  The  end  ele- 
vation of  the  gear-teeth  should  be  made  by  projecting  only 
the  points  of  the  teeth,  as  shown  at  No.  2. 


CONVENTIONAL    METHODS    OF    SHOWING    SCREW-THREADS  IN 
WORKING    DRAWINGS. 

FlG.  103. — No.  I,  shows  the  convention  for  a  double 
V  thread,  U.  S.  standard;'  No.  2,  a  single  V  thread;  No.  3, 
a  single  square  thread;  No.  4,  a  single  left-hand  V  thread; 
No.  5,  a  double  right-hand  square  thread;  No.  6,  any 
thread  of  small  diameter;  No.  7,  any  thread  of  very  small 
diameter.  The  true  methods  for  constructing  these  threads 
are  explained  on  pages  99-101,  Figs.  137-139. 

In  No.  6.  the  short  wide  line  is  equal  to  the  diameter 
of  the  thread  at  the  bottom.  The  distance  between  the 


CONVENTIONS. 


longer  narrow  lines  is  equal  to  the  pitch,  and  the  inclination 
is  equal  to  half  the  pitch. 

The  short  dash  lines   in   No.  7   should  be  made  to  corre- 


•   rti 


FIG.  103. 

spond  to  the  diameter  of  the  thread  at  the  bottom.  After 
some  practice  these  lines  can  be  drawn  accurately  enough  by 
the  eye. 


CHAPTER   IV. 
LETTERING  AND  FIGURING. 

THIS  subject  has  not  been  given  the  importance  it  deserves 
in  connection  with  mechanical  drawing.  Many  otherwise  ex- 
cellent drawings  and  designs  as  far  as  their  general  appearance 
is  concerned  have  been  spoiled  by  poor  lettering  and  figuring. 

All  lettering  on  mechanical  drawings  should  be  plain  and 
legible,  but  the  letters  in  a  title  or  the  figures  on  a  drawing 
should  never  be  so  large  as  to  make  them  appear  more  prom- 
inent than  the  drawing  itself. 

The  best  form  of  letter  for  practical  use  is  that  which  gives 
the  neatest  appearance  with  a  maximum  of  legibility  and  re- 
quires the  least  amount  of  time  and  labor  in  its  construction. 

This  would  naturally  suggest  a  "  free-hand  "  letter,  but  be- 
fore a  letter  can  be  constructed  "  free-hand  "  with  any  degree 
of  efficiency,  it  will  be  necessary  to  spend  considerable  time 
in  acquiring  a  knowledge  of  the  form  and  proportions  of  the 
particular  letter  selected. 

It  is  very  desirable  then  that  after  the  student  has  care- 
fully constructed  as  many  of  the  following  plates  of  letters  and 
numbers  as  time  will  permit  and  has  acquired  a  sufficient 
knowledge  of  the  form  and  proportions  of  at  least  the  "  Ro- 
man "  and  "  Gothic"  letters;  he  should  then  adopt  some  one 

64 


LETTERING  AND  FIGURING.  65 

style  and  practice  that  at  every  opportunity,  until  he  has  at- 
tained some  proficiency  in  its  free-hand  construction. 

When  practicing  the  making  of  letters  and  numbers  free- 
hand, they  should  be  made  quite  large  at  first  so  as  to  train 
the  hand. 

The  "  Roman  "  is  the  most  legible  letter  and  has  the  best 
appearance,  but  is  also  the  most  difficult  to  make  well,  either 
free-hand  or  mechanically.  However,  the  methods  given  for 
its  mechanical  construction,  Figs.  104  and  105,  will  materially 
modify  the  objections  to  its  adoption  for  lettering  mechanical 
drawings. 

The  "  Gothic"  letter  is  a  favorite  with  mechanical  drafts- 
men, because  it  is  plain  and  neat  and  comparatively  easy  to 
construct.  (See  Fig.  106.) 

Among  the  type  specimens  given  in  the  following  pages 
the  Bold-face  Roman  Italic  on  page  70  is  one  of  the  best 
for  a  good,  plain,  clear,  free-hand  letter,  and  is  often  used 
with  good  success  on  working  drawings.  Gillott's  No.  303 
steel  pen  is  the  best  to  use  when  making  this  letter  free-hand. 

The  "Yonkers"  is  a  style  of  letter  that  is  sometimes 
used  for  mechanical  drawings.  It  is  easy  to  construct  with 
either  F.  Soennecken's  Round  Writing-pens,  single  point,  or 
the  Automatic  Shading-pen.  But  it  lacks  legibility,  and  is 
therefore  not  a  universal  favorite. 

A  good  style  for  "  Notes"  on  a  drawing  is  the  "  Gothic 
Condensed  "  shown  on  page  70. 

When  making  notes  on  a  drawing  with  this  letter,  the 
only  guides  necessary  are  two  parallel  lines,  drawn  lightly  in 
pencil.  The  letters  should  be  sketched  lightly  in  pencil  first, 


66  MECHANICAL  DRAWING. 

and  then  carefully  inked,  improving  spacing  and  proportions 
to  satisfy  the  practiced  eye. 

FIGURING. 

Great  care  should  be  taken  in  figuring  or  dimensioning  a 
mechanical  drawing,  and  especially  a  working  drawing. 

To  have  a  drawing  accurately,  legibly,  and  neatly  figured 
is  considered  by  practical  men  to  be  the  most  important  part 
of  a  working  drawing. 

There  should  be  absolutely  no  doubt  whatever  about  the 
character  of  a  number  representing  a  dimension  on  a  drawing. 

Many  mistakes  have  been  made,  incurring  loss  in  time, 
labor,  and  money  through  a  wrong  reading  of  a  dimension. 

Drawings  should  be  so  fully  dimensioned  that  there  will 
be  no  need  for  the  pattern-maker  or  machinist  to  measure  any 
part  of  them.  Indeed,  means  are  taken  to  prevent  him  from 
doing  so,  because  of  the  liability  of  the  workman  to  make 
mistakes,  so  drawings  are  often  made  to  scales  which  are  dif- 
ficult to  measure  with  a  common  rule,  such  as  2 ''and  4"  = 
I  ft. 

The  following  books,  among  the  best  of  their  kind,  are 
recommended  to  all  who  desire  to  pursue  further  the  study 
of  "  Lettering"  :  Plain  Lettering,  by  Prof.  Henry  S.  Jacoby, 
Cornell  University,  Ithaca,  N.  Y.  ;  Lettering,  by  Charles  W. 
Reinhardt,  Chief  Draftsman,  Engineering  News,  New  York  ; 
Free-hand  Lettering,  by  F.  T.  Daniels,  instructor  in  C.  E.  in 
Tufts  College. 


LETTERING  AND  FIGURING. 


67 


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MECHANICAL   DRA  WING. 


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LETTERING  AND  FIGURING. 


70  MECHANICAL   DRAWING. 

i8-Point  Roman. 

ABCDEFGHIJKLMN  OPQRSTTJYWX 
YZ      abcdefghijklrmiopqrstuvwxyz 
1234567890 


i8-Point  Italic. 


AB  CDEFGHIJKLMNOP  QE8TUV 
WX  YZ    abcdefgMjklmnopqrs  tuviuxyz 

i?-Point  Gushing  Italic. 

ABCDEFGHIJKLMNOPQRSTUVWXYZ        abcdefghijklm 
nopqrstuvwxyz         1 23456  7890 


28-Point  Boldface  Italic. 


ABCDEFaHIJKLM 

NOPQRSTUVWXYZ 

dbcdefghijklmnopqrstu 

mvxyz    1234567890 

Two-Line  Nonpareil  Gothic  Gondensed. 

ABCDEFGHIJKLMNOPQRSTUVWXYZ      1234567890 

Three-Line  Nonpareil  Lightface  Celtic. 

ABCDEFGHIJKLMNOPQR 
STUVWXYZ        abedefghijkl 
mnopqrstuvwxyz 
1234567890 " 


LETTERING  AND  FIGURING.  Jl 

i8-Point  Chelsea  Circular. 

ABCDEFGHIJKLMNOPQRSTUVWX 
YZ     abcdefghiij^lmT)opqrstuvwxyz 
1234567890       '      . 

iS-Point  Elandkay. 

ABCDEFGHIJKLnNOFQRSTUVVXYZ 
1234567890 

1 8- Point  Quaint  Open. 


28-Point  Roman. 


ABCDEFGHIJKLM 
NOPQRSTUVWXYZ 

abcdefghij  klmnopqrstu 
vwxyz    1234567890 


28-Point  Old-Style  Italic. 


ABCDEFGHIJKLMNOP 

QRSTUVM/XYZ     abcdefg 
h  ijklm  n  opqrstuvwxyz 
1234567890 


MECHANICAL   DRAWING. 


i2-Point  Victoria  Italic. 

ABCDEFGHIJKLMNOPQRSTU 
VWXYZ      1234567890 


iS-Point  DeVinne   Italic. 


ABCDEFGHIJKLMNOPQRSTU 

VWX  YZ    abcdefghijklmnopqrst 

uvwxyz     1234567890 


22-Point  Gothic  Italic. 


ABCPEFGHIJKLMNOPQR8TUVWXYZ 

abcdefghijklmnopqrstuuwxyz 
1234567890 


Double- Pica  Program. 


ABCD: 


FGHIJKLMNO 
PQRSTUYWXYZ 
abcdefghijklmnopqrstuv 
wxyz    1234567890 


Nonpareil  Telescopic  Gothic. 

ABCDEFGHIJKLMNOPQRSTUVWXYZ          1234567S90 


LETTERING  AND  FIGURING. 


24-Point  Gallican. 


ABCDEFGHIJKL 
MNOPQRSTUVW 
XYZ  1234567890 


IIGI 


Two-Line  Virile  Open. 

p>/fi\ip; 
DlPi 


v: 


22-Point  Old-Style  Roman. 


ABCDEFGHIJKLMNOPQRST 

UVWXYZ       abcdefghijklmnopqrst 

uvwxyz      1234567890 


36-Point  Vonkers. 


^23^567890 


CHAPTER    V. 
ORTHOGRAPHIC   PROJECTION. 

ORTHOGRAPHIC  PROJECTION,  sometimes  called  Descrip- 
tive Geometry  and  sometimes  simply  Projection,  is  one  of 
the  divisions  of  descriptive  geometry;  the  other  divisions  are 
Spherical  Projection,  Isometric  Projection,  Shades  and 
Shadows,  and  Linear  Perspective. 

In  this  course  we  will  take  up  only  a  sufficient  number  of 
the  essential  principles  of  Orthographic  Projection,  Isometric 
Projection,  and  Shades  and  Shade  Lines,  to  enable  the  stu- 
dent to  make  a  correct  mechanical  drawing  of  a  machine  or 
other  object. 

Orthographic  Projection  is  the  science  and  the  art  of  rep- 
resenting objects  on  different  planes  at  right  angles  to  each 
other,  by  projecting  lines  from  the  po int  of  sight  through  the 
principal  points  of  the  object  perpendicular  to  the  Planes  of 
Projection. 

There  are  commonly  three  planes  of  projection  used,  viz., 
the  H.  P.  or  Horizontal  Plane,  the  V.  P.  or  Vertical  Plane, 
and  the  Pf.  P.  or  Profile  Plane. 

These  planes,  as  will  be  seen  by  Figs.  107  and  109,  inter- 
sect each  other  in  a  line  called  the  /.  L.  or  Intersecting  Line, 
and  form  four  angles,  known  as  the  first,  second,  third,  and 

74 


ORTHOGRAPHIC  PROJECTION. 


75 


fourth  Dihedral  Angles.  Figs.  107  and  109  are  perspective 
views  of  these  angles. 

An  object  may  be  situated  in  any  one  of  the  dihedral 
angles,  and  its  projections  drawn  on  the  corresponding  co- 
ordinate planes. 

Problems  in  Descriptive  Geometry  are  usually  worked  out 
in  the  first  angle,  and  nearly  all  English  draftsmen  project 
their  drawings  in  that  angle,  but  in  the  United  States  the 
third  angle  is  used  almost  exclusively.  There  is  good  reason 
for  doing  so,  as  will  be  shown  hereafter. 

We  will  consider  first  a  few  projection  problems  in  the 
first  angle,  after  which  the  third  angle  will  be  used  throughout. 


FIG.  107. 

H.P.,  Fig.  107,  is  the  Horizontal  Plane,  V.P.  the  Vertical 
Plane,  and  I.L.  the  Intersecting  Line. 

The  Horizontal  Projection  of  a  point  is  where  a  perpen- 
dicular line  drawn  through  the  point  pierces  the  H.P. 

The  Vertical  Projection  of  a  point  is  where  a  per.  line 
drawn  through  the  point  pierces  the  V.P. 

Conceive  the  point  <?,  Fig.  107,  to  be  situated  in  space  4" 
above  the  H.P.  and  3"  in  front  of  the  V.P.  If  a  line  is 
passed  through  the  point  a  per.  to  H.P.  and  produced  until 


76  *  MECHANICAL  DRAWING. 


it  pierces  the  H.P.  in  the  point  ak,  rt'will  be  the  Hor.  Proj. 
of  the  point  a. 

If  another  line  is  projected  through  the  points  per.  to  the 
V.P.  until  it  pierces  the  V.P.  in  the  point  av,  av  is  the  ver- 
tical projection  of  the  point  a. 

If  now  the  V.P.  is  revolved  upon  its  axis  I.L.  in  the  di- 
rection of  the  arrow  until  it  coincides  with  the  H.P.  and  let 
the  H.P.  be  conceived  to  coincide  with  the  plane  of  the 
drawing-paper,  the  projections  of  the  point  a  will  appear  as 
shown  by  Fig.  108. 

The  vertical  projection  a°  4"  above  the  I.L.  and  the 
horizontal  projection  ah  3"  below  the  I.L.  both  in  the  same 
straight  line. 

In  mechanical  drawing  the  vertical  projection  av  is  called 
the  Elevation  and  the  horizontal  projection  ah  the  Plan. 

The  projections  of  a  line  are  found  in  a  similar  manner, 
by  first  finding  the  projections  of  the  two  ends  of  the  line, 
and  joining  them  with  a  straight  line. 

Let  ab  be  a  line  in  space  32"  long,  parallel  to  the  V.P. 
and  perpendicular  to  the  H.P.  One  end  is  resting  on  the 
H.P.  2£"  from  the  V.P. 

The  points  a  and  b  will  be  vertically  projected  in  the 
points  a"  and  bv  .  Join  a°bv.  avbv  is  the  vertical  projection  of 
the  line  ab. 

When  a  line  is  perpendicular  to  one  of  the  planes  of  pro- 
jection, its  projection  on  that  plane  is  a  point,  and  the  projec- 
tion on  the  other  plane  is  a  line  equal  to  the  line  itself. 

ab,  Fig.  107,  is  perpendicular  to  the  H.P.,  therefore  its 
proj.  on  the  H.P.  when  viewed  in  the  direction  ab  will  be 
seen  to  be  a  point. 


ORTHOGRAPHIC  PROJECTION. 


77 


Conceive  now  the  V.P.  revolved  as  before,  the  V.  proj. 
will  be  found  to  be  at  a°bv,  Fig.  108,  and  the  H.  proj.  at  the 
point  ah. 

cd,  Fig.  107,  is  a  line  parallel  to  the  H.P.  and  perpendic- 
ular to  the  V.P.  Its  elevation  or  V.  proj.  is  the  point  dv,  Fig. 
108,  and  its  plan  or  H.  proj.  the  line  ckdh  perpendicular  to 
the  Intersecting  Line  and  equal  in  size  to  the  line  itself. 

Planes  or  Plane  Surfaces  bounded  by  lines  are  projected 
by  the  same  principles  used  to  project  lines  and  points. 

Let  aavbvb,  Fig.  107,  be  a  plane  at  right  angles  to  and 
touching  both  planes  of  projection. 

The  elevation  of  the  front  upper  corner  a  is  projected  in 
the  point  a*.  The  elevation  of  the  front  lower  corner  b  is  pro- 
jected in  the  point  bv.  Join  a°bv.  d°b°  is  the  vertical  projection 
of  the  front  edge  ab  of  the  plane.  The  plan  of  the  front 


FIG.  108. 

upper  corner  is  projected  in  the  point  b  and  the  point  0'in'the 
point  b°.  A  straight  line  joining  bbv  is  the  plan  or  horizontal 
projection  of  the  top  edge  of  the  plane. 

On  the  drawing-paper  the  plan  and  elevation  of  the  plane 
aavb  a  would  be  shown  as  a  continuous  straight  line  av  to  ah 
Fig.  108. 


78  MECHANICAL   DRAWING. 

Solids  bounded  by  plane  surfaces  are  projected  by  means 
of  the-same  principles  used  to  project  planes,  lines,  and  points. 

C,  Fig.  107,  is  a  cube  bounded  by  six  equal  sides  or  sur- 
faces. The  top  and  bottom  being  parallel  to  the  H.P.  and 
the  front  and  back  parallel  to  the  V.P.,  the  vert.  proj.  is  a 
square  above  I.L.  equal  in  area  to  any  one  of  the  six  faces 
of  the  cube.  The  hor.  proj.  is  a  similar  square  below  I.L. 

These   projections  are  shown  at  C,  Fig.   108,  as  they  would 
appear  on  the  drawing-paper. 

The  foregoing  illustrates  a  few  of  the  simple  principles  of 
projection  in  relation  to  points,  lines,  and  solids  when  placed 
in  the  first  dihedral  angle,  and  we  find  that  the  plan  is  always 
below  and  the  elevation  always  above  the  I.L. 

Let  us  now  consider  the  same  problems  when  situated  in 
the  third  angle.  The  points,  Fig.  109,  is  behind  of  the  V.P. 


FIG.  109. 

and  below  the  H.P.  Draw  through  a  perpendiculars  to  the 
plane  of  projection.  The  Hor.  proj.  is  found  at  ah  and  the 
vert.  proj.  at  a". 

Conceive  again  the  V.P.  to  be  revolved  in  the  direction 
of  the  arrow  until  it  coincides  with  the  H.  P.    The  hor.  proj. 


ORTHOGRAPHIC  PROJECTION. 


79 


will  then  appear  at  ah  above  the  I.L.  and  the  vert.  proj.  at  d° 
below  the  I.L.,  Fig.  no.  And  so.  with  the  lines,  the  planes, 
and  the  solids. 


FIG.  no. 


In  order  to  still  further  explain  the  use  of  the  planes  of 
projection,  with  regard  to  objects  placed  in  the  third  angle, 
let  us  suppose  a  truncated  pyramid  surrounded  by  imaginary 
planes  at  right  angles  to  each  other,  as  shown  by  Fig.  1 1 1 . 


FIG.  in. 

With  a  little  attention  it  will  easily  be  discerned  that  the 
pyramid  is  situated  in  the  third  dihedral  angle,  and  that  in 
addition  to  the  V.  and  H.  planes,  we  have  passed  two  profile 
planes  at  right  angles  to  the  V.  and  H.  planes,  one  at  the  right- 
hand  and  one  at  the  left. 

When  the  pyramid  is  viewed  orthographically  through 
each  of  the  surrounding  planes,  four  separate  views  are  had, 


8o 


MECHANICAL   DRA  WING. 


exactly  as  shown  by  the  projections  on  the  opposite  planes, 
viz.,  a  Front  View,  Elevation,  or  Vert.  Proj.  at  F.  ;  a  Right- 
hand  View,  Right-end  Elevation,  or  Right-profile  Projection 
at  R.;  a  Left-hand  View,  Left-end  Elevation,  or  Left-profile 
Projection  at  L.  ;  a  Top  View,  Plan  or  H.  Proj.  at  P. 

If  we  now  consider  the  V.P.  and  the  right  and  left  profile 
planes  to  be  revolved  toward  the  beholder  until  they  coincide, 
using  the  front  intersecting  lines  as  axes,  the  projections  of  the 
pyramid  will  be  seen  as  shown  by  Fiig.  112,  which  when  the 


\  / 


FIG.  112. 

imaginary  planes  and  projecting  lines  have  been  removed,  will 
be  a  True  Drawing  or  Orthographic  Projection  of  the  truncated 
pyramid. 

NOTATION. 

In  the  drawings  illustrating  the  following  problems  and 
their  solutions  the  given  andjTY 'quired  lines  are  shown  wide  and 
black.  Hidden  lines  are  shown  broken  into  short  dashes  a  little 
narrower  than  the  visible  lines.  Construction  or  projection  lines 
are  drawn  with  very  narrow  full  or  continuous  black  lines. 


ORTHOGRAPHIC  PROJECTION.  8 1 

When  convenient  very  narrow,  continuous  blue  lines  are  some- 
times used. 

The  Horizontal  Plane  is  known  as  the  H.P.,  the  Vertical 
Plane  as  V.P.  and  the  Profile  Plane  as  Pf.P. 

A  point  in  space  is  designated  by  a  small  letter  or  figure, 
their  projection  by  the  same  letters  or  figures  with  small  h  or 
v  written  above  for  the  horizontal  or  vertical  projection  re- 
spectively. 

In  some  complicated  problems  where  points  are  designated 
by  figures  their  projections  are  named  by  the  same  figures 
accented. 

Drawings  should  be  carefully  made  to  the  dimensions 
given,  the  scale  to  be  determined  by  the  instructor. 

The  student  should  continually  endeavor  to  improve  in 
inking  straight  lines,  curves,  and  joints. 

In  solving  the  following  problems  the  student  should  have 
a  model  of  the  co-ordinate  planes  for  his  own  use.  This  can 
be  made  by  taking  two  pieces  of  stiff  cardboard  and  cutting  a 
slot  in  the  center  of  one  of  them  large  enongh  to  pass  the 
folded  half  of  the  other  through  it ;  when  unfolding  this  half  a 
model  will  be  had  like  that  shown  by  Fig.  107  or  109. 

All  projections  shall  now  be  made  from  the  third, 
dihedral  angle. 

PROB.  i. — A  point  a  is  situated  in  the  third  dihedral 
angle,  i"  below  the  H.P.  and  3"  behind  the  V.P. 

It  is  required  to  draw  its  vertical  and  horizontal  projec- 
tions. 

Draw  a  straight  line  ahav,  Fig.  113,  perpendicular  to  I.L. 
and  measure  off  the  point  a"  i"  below  I.L.  and  the  point  ah 
3"  above  I.L. 


82 


MECHANICAL   DRAWING. 


d"  is  the  vertical  and  ah  the  horizontal  projection  in  the 
same  straight  line  a"ah. 

The  student  should  demonstrate  this  with  his  model. 

PROB.  2. — Draw  two  projections  of  a  line  3"  long  parallel 
to  both  planes,  I"  below  the  H.P.  and  2"  behind  the  V.P. 

As  the  line  is  parallel  to  both  planes,  both  projections  will 
be  parallel  to  the  I.L. 

Draw  a"^  the  vert.  proj.  of  the  line  3"  long,  Fig.  1 14,  par- 
allel to  I.L.  and  j-"  below  it.  Draw  the  hor.  proj.  2"  above 
the  I.L.  and  parallel  to  it,  making  it  the  same  length  as  the 


FIG.  113.       FIG.  114.     FIG.  115.      FIG.  116.  FIG.  117. 

vert.  proj.  by  drawing  lines  perpendicular  to  I.L.  from  the 
points  a"  and  bv  to  ah  and  //. 

PROB.  3.-^To  draw  the  hor.  and  vert,  projs.  of  a  straight 
line  3"  long,  per.  to  the  vert,  plane,  Fig.  115. 

As  the  line  is  per.  to  the  vert,  plane  the  vert.  proj.  will  be 
a  point  below  the  I.L.  and  the  hor.  proj.  will  be  parallel  to 
the  horizontal  plane  and  per.  to  I.L. 

PROB.  4. — To  draw  the  plan  and  elevation  of  a  straight 
line  6"  long  making  an  angle  of  45°  with  the  vert,  plane  and 
and  par.  to  the  hor.  plane,  Fig.  116. 


ORTHOGRAPHIC  PROJECTION.  83 

The  plan  or  hor.  proj.  will  be  above  the  I.L.  and  make  an 
angle  of  45°  with  it.  The  elevation  or  vert.  proj.  will  be 
below  and  par.  to  I.L. 

Draw  from  the  point  ah  at  any  convenient  distance  from 
I.L.  a  straight  line  ahbh  6"  long,  making  an  angle  45°  with  I.L. 

Draw  a°bv  par.  to  I.L.  at  a  convenient  distance  below  it. 
The  length  of  the  elevation  or  vert.  proj.  is  determined  by 
dropping  perpendiculars  from  the  end  of  the  hor.  proj.  o*^*to 
the  points  a"bv. 

PROB.  5,  FlG.  117. — To  find  the  true  length  of  a  straight 
line  oblique  to  both  planes  of  projection  and  the  angle  it 
makes  with  these  planes. 

a*b*  and  ahbh  are  the  projections  of  a  straight  line  oblique 
to  V.P.  and  H.P.  Using  av  as  a  pivot,  revolve  the  line  d°bv 
until  it  becomes  parallel  to  I.L.  as  shown  by  avb*.  From  the 
point  b?  erect  a  per.  Through  the  point  bh  draw  a  line  par.  to 
I.L.  cutting  the  per.  in  the  point  b*. 

The  broken  line  c^bf  is  the  true  length  of  the  line  abt 
and  the  angle  o  is  the  true  angle  which  the  line  makes  with 
V.P. 

To  find  the  angle  it  makes  with  H.P.  : 

Using  tf1  as  a  pivot,  revolve  the  line  bkah  until  it  becomes 
par.  to  I.L.  as  shown  by  bha*.  From  the  point  a?  drop  a  per. 
Through  the  point  a"  draw  a  line  par.  to  I.L.  intersecting  tke 
per.  at  the  point  a?o  is  the  angle  which  the  line  ab  makes 
with  H.P.  and  the  broken  line  a?b°  is  again  its  true  length. 

PROB.  6,  FlG.  118. — To  project  a  plane  surface  of  given 
size,  situated  in  the  third  angle  and  par.  to  the  V.P. 

Let  abed  be  the  plane  surface  3"  long  X  2"  wide.  If 
we  conceive  lines  to  be  projected  from  the  four  corners  of  the 


84  MECHANICAL   DRAWING. 

plane  surface  to  the  V.P.  and  join  them  with  straight  lines  we 
will  have  its  V.  projection  avl?vcvdv  and  shown  by  Fig.  118. 
And  as  the  plane  surface  is  par.  to  the  V.P.  it  must  be  per 
to  the  H.P.  since  the  planes  of  projection  are  at  right  angles 
to  each  other.  So  the  plan  or  H.  projection  will  be  a  straight 
line  equal  in  length  to  one  of  the  sides  of  the  plane  surface. 

At  a  convenient  distance  above  I.L.  draw  a  straight  line, 
and  from  the  points  avbv  project  lines  at  right  angles  to  I.L., 
cutting  the  straight  line  in  the  points  ahb.h  The  line  ahbh  is 
the  hor.  proj.  of  the  plane  surface  abed. 

PROB.  7,  FIG.  118. — To  draw  the  projections  of  a  plane 
surface  of  given  dimensions  when  situated  in  the  third  angle 
perpendicular  to  the  H.P.  and  making  an  angle  with  the  V.P. 

Let  the  plane  surface  be  3"  X  2"  as  before  and  let  the 
angle  it  makes  with  V.P.  be  60°. 

To  draw  the  plan : 

At  a  convenient  distance  above  I.L.  and  making  an  angle 
of  60°  with  it,  draw  ahbf,  Fig.  1 18,  2"  long.  From  /;/'  drop  a 
per.  cutting  a"bv  in  the  point  b?  and  c°dv  in  the  point  d? ,  then 
the  rectangle  avb?d?cv  will  be  the  vert.  proj.  or  elevation  of 
the  plane  surface  abed. 

PROB.  8,  FIG.  119. — To  draw  the  projections  of  the  same 
plane  surface  (i)  when  parallel  to  the  H.P.,  (2)  when  making 
an  angle  of  30°  with  H.P.  and  per.  to  V.P.,  (3)  when  mak- 
ing an  angle  of  60°  with  H.P.  and  per.  to  V.P.,  and  (4)  when 
per.  to  both  planes. 

Fig.  119  shows  the  projections;  further  explanations  are 
unnecessary. 

PROB.  9,  FIGS.  1 19  AND  120. — To  draw  the  projections  of 


ORTHOGRAPHIC  PROJECTION. 


the  same  plane  surface  when  making  compound  angles  with 
the  planes  of  projection. 

Let  the  plane  make  an  angle  of  30°  with  H.P.,  as  in  the 
second  position  of  Prob.  8,  Fig.  119,  and  in  addition  to  that, 
revolve  it  through  at  angle  of  30°.  First,  draw  the  plane 
parallel  to  H.P.,  as  shown  by  ahchbkdk,  Fig.  1 19,  the  true  size 
of  the  plane. 


!*"<"-.-• 

FIG.  118.  FIG.  119.  FIG.  120. 

Its  elevation  will  be  the  straight  line  avb"  parallel  to  I.L. 
Next  revolve  avbv,  using  a''  as  a  pivot,  through  an  angle  of 
30°,  to  the  position  a°b*,  which  is  its  vert.  proj.  when  making 
an  angle  of  30°  with  H.P.  Its  plan  is  projected  in  ahbf<?d*. 

Now  as  the  plane  is  still  to  make  an  angle  of  30°  with 
H.P.  after  it  has  been  revolved  through  an  angle  of  30°  with 
relation  to  the  V.P.,  its  hor.  proj.  will  remain  unchanged. 

With  a  piece  of  celluloid  or  tracing-paper  trace  the  hor. 
proj.  rt^V*/,*,  lettering  the  points  as  shown,  and  revolve  the 


86  MECHANICAL   DRAWING. 

tracing  through  the  angle  of  30°,  or,  which  is  the  same  thing, 
place  the  tracing  so  that  the  line  ahch  will  make  an  angle  of 
60°  with  I.L.,  and  with  a  sharp  conical-pointed  pencil  trans- 
fer the  four  points  to  the  drawing-paper  and. join  them  by 
straight  lines,  as  shown  by  Fig.  120. 

And  as  the  line  ahch  retains  its  position  relative  to  H.P. 
after  the  revolution,  its  elevation  will  be  found  at  avcv,  Fig. 
120,  in  a  straight  line  drawn  through  cfb",  Fig.  119,  intersect- 
ing perpendiculars  from  ahch,  Fig.  120.  And  the  vert.  proj. 
of  the  points  bfdf  will  be  found  at  /v'^7',  Fig.  120,  in  a  straight 
line  drawn  through  b*,  Fig.  1 19,  parallel  to  I.L.  and  intersect- 
ing pers.  from  #,V/*,  join  with  straight  lines  the  points 
cPb*ed?. 

Draw  the  projections  of  the  plane  when  making  an  angle 
of  60°  with  H.P.  and  revolved  through  an  angle  of  30°  with 
relation  to  V.P. 

Draw  the  projections  of  the  plane  when  making  an  angle 
of  60°  with  the  V.P.  and  per.  to  the  H.P.,  Fig.  120. 

PROB.  10. — To  draw  the  projections  of  a  plane  surface  of 
hexagonal  form  in  the  following  positions:  (i)  When  one 
of  its  diagonals  is  par.  to  the  V.P.  and  making  an  angle  of 
45°  with  the  H.P.  (2)  When  still  making  an  angle  of  45° 
with  the  H.P.  the  same  diagonal  has  been  revolved  through, 
an  angle  of  60°. 

Draw  the  hexagon  ih2h$ &4*5A6A,  Fig.  121,  at  any  Con- 
venient distance  above  I.L.,  making  the  inscribed  circle 
—  2J-".  This  will  be  its  hor.  proj.  and  2v^v6viv  its  vert,  proj., 
the  diagonal  ih2h  being  par.  to  both  planes  of  proj.  With 
I*  as  an  axis  revolve  6V4V2V  through  an  angle  of  45°.  Through 
the  points  2*4*6*  erect  pers.  to  the  points  6l*51A4I*3,A  and  2,A 


ORTHOGRAPHIC  PROJECTION.  8/ 

and  join  them  with  straight  lines.  These  are  the  projs.  in 
the  first  position.  Now  trace  the  hor.  proj,  IA,  2,A,  etc.,  on 
a  piece  of  celluloid  or  tracing-paper  and  revolve  the  tracing 
until  the  diagonal  iA2aA  makes  an  angle  of  60°  with  the  I.L., 
Fig.  122.  Next  draw  pers.  from  the  6  points  of  the  hexag- 
onal plane  to  intersect  hors.  from  the  corresponding  points  of 
the  elevation  in  Fig.  121,  join  the  points  of  intersection  with 


straight  lines,  and  so  complete  the  projections  of  the  second 
position,  Fig.  122. 

PROB.  n,  FIGS.  123  AND  124. — Draw  the  projs.  of  a  cir- 
cular plane  (i)  when  its  surface  is  par.  to  the  vert,  plane,  (2) 
when  it  makes  an  angle  of  45°  with  the  V.P.,  and  (3)  when 
still  making  an  angle  of  45°  with  the  V.P.  it  has  been  re- 
volved through  an  angle  of  60°. 

First  position:  Draw  the  circular  plane  r,  2V,  3",  4*,  etc., 
Fig.  123,  below  the  I.L.  with  a  radius  =  i  J"  and  divide  and 
figure  it  as  shown. 


88 


MECHANICAL   DRAWING. 


Since  the  plane  is  par.  to  V.P.  its  hor.  proj.  will  be  a 
straight  line  i;',  2/f,  .....  etc. 

For  the  second  position  revolve  the  said  hor.  proj.  through 
the  required  angle  0(45°  to  the  position  ah  .  .  .  .  I/',  Fig.  123, 
and  through  each  division  in  ik  .  .  .  .  ah  draw  arcs  cutting 
ah  .  .  .  .  IA  in  points  2h^h  .  .  .  This  is  the  hor.  proj.  of  the 
plane  when  making  an  angle  of  45°  with  the  V.P. 

The  elevation  is  found  by  dropping  pers.  from  the  points 
in  the  hor.  proj.  «*...!,*  to  intersect  hor.  lines  drawn 
through  the  correspondingly  numbered  points  in  the  eleva- 


FIG.  123. 


tion  and   through   these   intersections  draw   the  elevation   or 
vert.  proj.  of  the  second  position. 

For  the  third  position  make  a  tracing  of  the  elevation  of 
the  second  position,  numbering  all  the  points  as  before,  and 
place  the  tracing  so  that  the  diameter  7^7"  makes  the  required 
angle  of  60°  with  the  I.L.  and  transfer  to  the  drawing-paper. 


ORTHOGRAPHIC  PROJECTION.  89 

The  result  will  be  the  elevation  of  the  third  position  shown 
below  the  I.L.,  Fig.  124.  Its  hor.  proj.  is  found  by  drawing 
pers.  through  the  points  I,  2,  3,4  ...  to  intersect  hors.  drawn 
through  the  corresponding  points  in  the  hor.  proj.  of  the  2d 
position  and  through  these  intersections  draw  the  plan  or  hor. 
proj.  of  the  third  position,  Fig.  124. 

PROB.  12,  FIG.  125. — Draw  the  projs.  of  a  regular  hexag- 
onal prism,  3"  high  and  having  an  inscribed  circle  of  4f"' 
diam.  :  (i)  When  its  axis  is  par.  to  the  V.P.  (2)  Draw  the 
true  form  of  a  section  of  the  prism  when  cut  by  a  plane 
passing  through  it  at  an  angle  of  30°  with  its  base.  (3) 
Draw  the  projection  of  a  section  when  cut  by  a  plane  passing 
through  XX,  Fig.  125,  per.  to  both  planes  of  proj. 

The  drawing  of  the  I.L.  may  now  be  omitted. 

For  the  plan  of  the  first  part  of  this  prob.  draw  a  circle 
with  a  radius  =  to  2T5¥",  and  circumscribe  a  hexagon  about  it, 
as  shown  by  ah,  bh,  If,  etc.,  Fig.  125.  To  project  the  elevation, 
draw  at  a  convenient  distance  from  the  plan  a  hor.  line  par. 
to  ahd:\  and  3"  below  it  another  line  par.  to  it.  From  the 
points  ahbh^ldh,  drop  pers.  cutting  these  par.  lines  in  the  points 
avbvcvdv,  thus  completing  the  elevation  of  the  prism. 

Second  condition :  Draw  the  edge  view  or  trace  of  the 
cutting  plane  i'4'»  making  an  angle  of  30°  with  the  base  of  the 
prism,  locating  the  lower  end  4'  one-half  inch  above  the  base; 
parallel  to  iV>  and  at  a  convenient  distance  from  it  draw  a 
straight  line  1,4;  at  a  distance  of  2-jj>-$"  on  each  side  of  1,4 
draw  lines  3,  2  and  5,  6  parallel  to  1,4,  and  through  the 
points  x'2'3'4'  let  fall  pers.  cutting  these  three  par.  lines  in 
the  points  I,  2,  3,  4,  5,  6;  join  these  points  by  straight  lines 


9o 


ME  CHA  NICAL   D  RA  WING . 


as  shown,  and  a  true  drawing  of  the  section  of  the  prism  as 
required  will  result. 

For  the  third  condition  of  the  problem  : 

Let  XX  be  the  edge  view  of  the  cutting  plane  and  con- 
ceive that  part  of  the  prism  to  the  right  of  XX  to  be  removed. 


FIG.  125.  FIG.  126. 

From  the  hor.  proj.  of  the  prism  draw  a  right-hand  elevation 
or  profile  proj.,  and  through  the  points  XX  draw  the  lines  en- 
closing the  section,  and  hatch-line  it  as  shown. 

PROB.    13. — To  draw  the  development  of  the  lower  part 
of  the  prism  in  the  elevation  of  the  last  problem. 


• 


ORTHOGRAPHIC  PROJECTION.  91 

To  the  right  of  the  elevation  in  Fig.  125,  prolong  the  base 
line  indefinitely  and  lay  off  upon  it  the  distances  ab,  be,  cd, 
etc.,  Fig.  126,  each  equal  in  length  to  a  side  of  the  hex.  At 
these  points  erect  pers.,  and  through  the  points  i'2r^'^  draw 
hor.  lines  intersecting  the  pers.  in  4,  3,  2,  I,  etc.  At  be 
draw  the  hex.  ahbkbkjckchj^  of  the  last  prob.  for  the  base,  and 
at  I,  2  draw  the  section  I,  2,  3,  4,  5,  6  for  the  top. 

PROB.  14,  FlG.  127. — To  draw  the  projs.  of  a  right  cylin- 
der 3"  diam.  and  3''  long,  (i)  When  its  axis  is  per.  to  the 
H.P.  (2)  Draw  the  true  form  of  a  section  of  the  cylinder, 
when  cut  by  a  plane  per.  to  the  V.P.  making  an  angle  of  30° 
with  the  H.P.  (3)  Draw  a  development  of  the  upper  part  of 
the  cyl. 

For  the  plan  of  the  first  condition,  describe  the  circle  i', 
2',  etc.,  with  a  radius  =  \\"  and  from  it  project  the  eleva- 
tion, which  will  be  a  square  of  3"  sides. 

For  the  second  condition:  Let  i,  7  be  the  trace  of  the 
cutting  plane,  making  the  point  7,  y  from  the  top  of  the  cyl. 
Divide  the  circle  into  12  equal  parts  and  let  fall  pers.  through 
these  divisions  to  the  line  of  section,  cutting  it  in  the  points 
i,  2,  3,4,  etc.  Parallel  to  the  line  of  section  i,  7  draw  i"f 
at  a  convenient  distance  from  it,  and  through  the  points 
i,  2,  3,  4,  etc.,  draw  pers.  to  1,7,  intersecting  and  extending 
beyond  \"f.  Lay  off  on  these  pers.  the  distances  6' '8"  — 
6'8',  and  $"9"  =  $'$',  etc.,  and  through  the  points  2",  3", 
4".  etc.,  describe  the  ellipse. 

For  the  development:  In  line  with  the  top  of  the  eleva- 
tion draw  the  line  g'g"  equal  in  length  to  the  circumference  of 
the  circle,  and  divide  it  into  12  equal  parts  a',  b' ,  etc.,  a' ,  b" ', 
etc.  Through  these  points  drop  pers.  and  through  the  points 


92 


MECHANICAL   DRA  WING. 


I,  2,  3,  etc.,  draw  hors.  intersecting  the  pers.  in  the  points 
I,  2,  3,  etc.,  and  through  these  points  draw  a  curve. 

Tangent  to  any  point  on  the  straight  line  draw  a  3"  circle 
for  the  top  of  the  cyl.  and  tangent  to  any  suitable  point  on 
the  curve  transfer  a  tracing  of  the  ellipse. 

PROB.  15,  FlG.  128. — Draw  the  projections  of  a  right  cone 
7"  high,  with  a  base  6"  in  diam.,  pierced  by  aright  cyl.  2"  in 


gfedcbab 


FIG.  127. 

diam.  and  5"  long  their  axes  intersecting  at  right  angles  3" 
above  the  base  of  the  cone  and  par.  to  V.P.  Draw  first  the 
plan  of  the  cone  with  a  radius  =  3". 

At  a  convenient  distance  below  the  plan  draw  the  elevation 
to  the  dimensions  required. 

3"  above  the  base  of  the  cone  draw  the  center  line  of  the 
cyl.  CD,  and  about  it  construct  the  elevation  of  the  cyl.,  which 
will  appear  as  a  rectangle  2"  wide  and  2\"  each  side  of  the 
axis  of  the  cone.  The  half  only  appears  in  the  figure. 


CM  THOGRAPHIC  PROJECTION. 


93 


To  project  the  curves  of  intersection  between  the  cyl.  and 
cone  in  the  plan  and  elevation :  Draw  to  the  right  of  the  cyl. 
on  the  same  center  line  a  semicircle  with  a  radius  equal  that 
of  the  cyl.  Divide  the  semicircle  into  any  number  of  parts, 


FIG.  128. 


FIG.  129. 


as  i,  2,  3,  4,  etc.  Through  I,  I  draw  the  per.  A"  \"  equal 
in  length  to  the  height  of  the  cone,  and  through  A"  draw  the 
line^'V'  tangent  to  the  semicircle  at  the  point  4,  and  through 
the  other  divisions  of  the  semicircle  draw  lines  from  A"  to  the 
line  i'V'>  meeting  it  in  the  points  3  "2". 

From   all    points  on   the  line    i"4''»  vlz-«  i"2"3"4''»  erect 


94  MECHANICAL   DRAWING. 

pers.  to  the  center  line  of  the  plan,  cutting  it  in  the  points 
ii"2i"3i"4i"»  and  with  I,"  as  the  center  draw  the  arcs  2,"-2, 
3, "-3,  4i"-4  above  the  center  line  of  the  plan,  and  through  the 
points  2,  3,  4  draw  hors.  to  intersect  the  circle  of  the  plan  in 
the  points  2/3/4/,  and  lay  off  the  same  distances  on  the  other 
side  of  the  center  line  of  the  plan  in  same  order,  viz.,  2/3/4/. 
Through  each  of  these  points  on  the  circumference  of  the  circle 
of  the  plan  draw  radii  to  its  center  A',  and  through  the  same 
points  also  in  the  plan  let  fall  pers.  to  the  base  of  the  elevation 
of  the  cone,  cutting  it  in  the  points  2/3/4/ ;  and  from  the  apex 
A  of  the  elevation  of  the  cone  draw  lines  to  the  points  2^4'  on 
the  base.  Hor.  lines  drawn  through  the  points  of  division  2, 
3,  4  on  the  semicircle  will  intersect  the  elements  A— 2' ,  A—^'j 
A-4  of  the  cone  in  the  points  2/3/4/;  these  will  be  points  in 
the  elevation  of  the  curve  of  intersection  between  the  cylinder 
and  the  cone. 

The  plan  of  the  curve  is  found  by  erecting  pers.  through 
the  points  in  the  elevation  of  the  curve  to  intersect  the  radial 
lines  of  the  plan  in  correspondingly  figured  points,  through 
which  trace  the  curve  as  shown.  Repeat  for  the  other  half 
of  the  curve. 

PROB.  16,  FIG.  129. — To  draw  the  development  of  the 
half  cone,  showing  the  hole  penetrated  by  the  cyl. 

With  center  4,",  Fig.  129,  and  element  A\'  of  the  cone, 
Fig.  128,  as  radius,  describe  an  arc  equal  in  length  to  the  semi- 
circle of  the  base  of  the  cone.  Bisect  it  in  the  Iine4,"i,  and 
on  each  side  of  the  point  I  lay  off  the  distances  2,  3,  4,  equal 
to  the  divisions  of  the  arc  in  the  plan  Fig.  128,  and  from  these 
points  draw  lines  to  4",  the  center  of  the  arc.  Then  with 
radii  A-a,  b,  c,  d,  e,  respectively,  on  the  elevation  Fig.  128, 


ORTHOGRAPHIC  PROJECTION. 


95 


and  center  4,"  draw  arcs  intersecting  the  lines  drawn  from  the 
arc  XX  to  its  center  4,".  Through  the  points  of  intersection 
draw  the  curve  as  shown  by  Fig.  129. 

PROB.  17,  FlG.  130. — To  draw  the  development  of  the 
half  of  a  truncated  cone,  given  the  plan  and  elevation  of 
the  cone. 


FIG.  130. 

Divide  the  semicircle  of  the  plan  into  any  number  of  parts, 
then  with  A  as  center  and  A  I  as  radius,  draw  an  arc  and  lay 
off  upon  it  from  the  point  I  the  divisions  of  the  semicircle 
from  i  to  9,  draw  gA.  Then  with  center  A  and  radius  AB 
draw  the  arc  BC.  iBCg  is  the  development  of  the  half  of 
the  cone  approximately. 


op  MECHANICAL    DRA  WING. 

PROB.  1 8,  FIG.  131. — To  draw  the  curve  of  intersection  of 
a  small  cyl.  with  a  larger.  To  the  left  of  the  center-line  of 
Fig.  131  is  a  half  cross-section,  and  to  the  right  a  half  eleva- 
tion, of  the  two  cyls. 

Draw  the  half  plan  of  the  small  cyl.,  which  will  be  a 
semicircle,  and  divide  it  into  any  convenient  number  of  parts, 
say  12. 

From  each  of  these  divisions  drop  pers. 

On  the  half  cross-section  these  pers.  intersect  the  circum- 
ference of  the  large  cyl.  in  the  points  i',  2',  etc.  Through 


FIG.  134. 


FIG.  133 


FIG.  132. 

these  points  draw  hors.  to  intersect  in  corresponding  points 
the  pers.  on  the  half  elevation.  Through  the  latter  points 
draw  the  curve  of  intersection  C. 

PROB.  19. — To  draw  the  development  of  the  smaller  cyl. 
of  the  last  prob. 

Draw  a  rectangle,  Fig.  132,  with  sides  equal  to  the  circum- 


ORTHOGRAPHIC  PROJECTION.  97 

ference  and  length  of  the  cyl.  respectively,  and  divide  it  into 
24  equal  parts. 

Make  AB,  I  i',  3  3',  etc.,  Fig.  132,  equal  to  AB,  iV, 
2/2//,  3'3",  etc.,  Fig.  131,  and  draw  the  developed  curve  of 
intersection. 

PROB.  20. — To  draw  the  orthographic  projections  of  a 
cylindrical  dome  riveted  to  a  cylindrical  boiler  of-  given 
dimensions. 

Let  the  dimensions  of  the  dome  and  boiler  be.:  dome 
26J"  diam.  X  27"  high,  boiler  54"  diam.,  plates  £"  thick. 

Apply  to  the  solution  of  this  problem  the  principles  ex- 
plained in  Prob.  No.  18,  Fig.  131. 

When  your  drawings  are  completed,  compare  them  with 
Figs.  133  and  134,  which  are  the  projections  "required  in  the 
problem. 

Letter  or  number  the  drawing  and  be  prepared  to  explain 
how  the  different  projections  were  found. 

PROB.  21. — To  draw  the  development  of  the  top  gusset- 
sheets  of  a  locomotive  wagon-top  boiler  of  given  dimensions. 

First  draw  the  longitudinal  cross-section  of  the  boiler  to 
the  dimensions  given  by  Fig.  135,  using  the  scale  of  \"  = 
i  ft. 

Then  at  _any  convenient  point  on  your  paper  draw  a 
straight  Jine,,  _and  upon_it  jay  off  a  distance  AB  35^"  long  = 
the  straight  part  of  the  top  of  the  gusset-sheet  G,  Fig.  135. 
With  center  A  and  a  radius  =  27^"  (the  largest  radius  of  the 
gusset)  -f-  6"  (the  distance  from  the  center  of  the  boiler  to  the 
center  of  the  gusset  C,  Fig.  135)  =  33i",  draw  arc  I. 

With  center  B  and  a  radius  =  26£"  (the  smallest  radius  of 
the  gusset)  draw  arc  2.  Tangent  to  these  arcs  draw  the 


98 


MECHANICAL    DKA  WING. 


straight  line  I,  2  extended,  and  through  the  points^  and 
draw  lines  I,  A  and  2,  B  per.  to  I,  2. 


Take  a  point  on  the  per.  I,  2,  6"  from  the  point  I   as  a 
center  and  through  the  point   A   draw  an  arc  with  a  radius 


ORTHOGRAPHIC  PROJECTION.  99 

vVith  point  2  as  a  center  and  2B  as  a  radius  (26%")  draw 
an  arc  through  B  to  meet  the  line  1,2. 

Divide  both  arcs  into  any  number  of  parts,  say  12,  and 
through  these  divisions  draw  lines  per.  to  and  intersecting  \A 
and  2B  respectively.  Through  these  intersections  draw  in- 
definite hors.  and  on  these  hors.  step  off  the  length  of  the 
arcs,  with  a  distance  =  one  of  the  12  divisions  as  follows: 

On  the  first  hors.  lay  off  the  length  of  the  arc  Ai'  and  B\' 
=  Ai  and  Bi  respectively.  Then  from  T  lay  off  the  same 
distance  to  2'  on  the  second  hors.  etc.  Through  these  points 
draw  curves  Ai^'  and  Bi2f.  Join  points  12'  and  13'  with  a 
straight  line  Then  ABi2,  13  will  be  the  developed  half  of 
the  straight  part  of  the  gusset. 

On  the  two  ends  or  front  and  back  of  the  gusset  we  have 
now  to  add  i"  for  clearance  +  3f"  for  lap  +  \"  allowance 
for  truing  up  the  plates,  total  =  5J".  And  to  the  sides  2-f" 
for  lap  +  y  allowance  for  truing  up,  total  =  3i". 

The  outline  of  the  developed  sheet  may  now  be  drawn  to 
include  these  dimensions  with  as  little  waste  as  possible,  as 
shown  by  Fig.  136.  Extreme  accuracy  is  necessary  in  mak- 
ing this  drawing,  as  the  final  dimensions  must  be  found  by 
measurement. 

PROB.  22. — To  draw  the  projections  of  a  V-threaded 
screw  and  its  nut  of  3"  diam.  and  J"  pitch. 

Begin  by  drawing  the  center  line  C,  Fig.  137,  and  lay  off 
on  each  side  of  it  the  radius  of  the  screw  ij-".  Draw  AB 
and  6D.  Draw  A6  the  bottom  of  the  screw,  and  on  AB  step 
off  the  pitch  =  J",  beginning  at  the  point  A. 

On  line  6D  from  the  point  6  lay  off  a  distance  =  half  the 
pitch  =  f ",  because  when  the  point  of  the  thread  has  com- 


100 


ME  CHA  NICA  L   D  RA  WING. 


pleted  half  a  revolution  it  will  have  risen  perpendicularly  a 
distance  =  half  the  pitch,  viz.,  f". 

Then  from  the  point  6"  on  6D  step  off  as  many  pitches  as 
may  be  desired.  From  the  points  of  the  threads  just  found, 

B  D 


FIG.  137.  FIG.  138. 

draw  with  the  30°  triangle  and  T-square  the  V  of  the  threads 
intersecting  at  the  points  b  .  .  b  .  .  the  bottom  of  the  threads. 

At  the  point  O  on  line  A6  draw  two  semicircles  with  radii 

||   the  top  and   bottom   of  the  thread    respectively.      Divide 

these  into  any  number  of  equal  parts  and  also  the  pitch  Pinto 

the  same  number  of   equal  parts.      Through  these  divisions 

draw  hors.   and  pers.  intersecting  each  other  in  the  points  as 


OR  THOGRA PHIC  PROJECTION. 


101 


shown  by  Fig.  137,  which  shows  an  elevation  partly  in  section 
and  a  section  of  a  nut  to  fit  the  screw.  Through  the  points 
of  intersection  draw  the  curves  of  the  helices  shown,  using 
No.  3  of  the  4<  Sibley  College  Set"  of  Irregular  Curves. 


FIG.  139. 

PROB.  22. — To  draw  the  proj.  of  a  square-threaded  screw 
3"  diam.  and  \"  pitch  and  also  a  section  of  its  nut. 

The  method  of  construction  is  the  same  as  for  the  last 
problem,  and  is  illustrated  by  Fig.  138. 

PROB.  22. — To  draw  the  projections  of  a  square  double 
threaded  screw  of  3"  diam.  and  2"  pitch,  and  also  a  section  of 
its  nut. 


102 


MECHANICAL   DRA  WING. 


The  solution  of  this  problem  is  shown  by  Fig.  139,  and 
further  explanation  should  be  unnecessary. 

PROB.  23. — To  draw  the  curve  of  intersection  that  is 
formed  by  a  plane  cutting  an  irregular  surface  of  revolution. 


FIG.  140. 

Figs.  140,  141,    and    142   show  examples  of  engine  con- 
necting rod  ends  where  the  curve  /  is  formed  by  the  inter- 


FIG.  141. 

section  of  the  flat  stub  end  with 'the  surface  of  revolution  of 
the  turned  part  of  the  rod. 


ORTHOGRAPHIC  PROJECTION. 


103 


The  method  of  finding  the  curves  of  intersection  are  so 
plainly  shown  by  the  figures  that  a  detailed  explanation  is 
deemed  unnecessary. 


FIG.  142. 

SHADE    LINES,    SHADES   AND    SHADOWS. 

Shade  Lines  are  quite  generally  used  on  engineering  work- 
ing drawings;  they  give  a  relieving  appearance  to  the  projec- 
ting parts,  improve  the  looks  of  the  drawing  and  make  it  easier 
to  read,  and  are  quickly  and  easily  applied. 

The  Shading  of  the  curved  surfaces  of  machine  parts  is 
sometimes  practiced  on  specially  finished  drawings,  but  on 
working  drawings  most  employers  will  not  allow  shading  be- 
cause it  takes  too  much  time,  and  is  not  essential  to  a  quick 
and  correct  reading  of  a  drawing,  especially  if  a  system  of 
shade  lines  is  used. 

Some  of  the  principles  of  shade  lines  and  shading  are 
given  below,  with  a  few  problems  illustrating  their  commonest 
applications. 

The  shadows  which  opaque  objects  cast  on  the  planes  of 


104  MECHANICAL   DRAWING. 

projection  or  on  other  objects  are  seldom  or  never  shown  on 
a  working  drawing,  and  as  the  students  in  Sibley  College  are 
taught  this  subject  in  a  course  on  Descriptive  Geometry,  it  is 
omitted  here. 

CONVENTIONS. 

The  Source  of  Light  is  considered  to  be  at  an  infinite  dis- 
tance from  the  object,  therefore  the  Rays  of  Light  will  be  rep- 
resented by  parallel  lines. 

The  Source  of  Light  is  considered  to  be  fixed,  and  the  Point 
of  Sight  situated  in  front  of  the  object  and  at  an  infinite  dis- 
tance from  it,  so  that  the  Visual  Rays  are  parallel  to  one 
another  and  per.  to  the  plane  of  projection. 

Shade  Lines  divide  illuminated  surfaces  from  dark  surfaces. 

Dark  surfaces  are  not  necessarily  to  be  defined  by  those 
surfaces  which  are  darkened  by  the  shadow  cast  by  another 
part  of  the  object,  but  by  reason  of  their  location  in  relation 
to  the  rays  of  light. 

It  is  the  general  practice  to  shade-line  the  different  pro- 
jections of  an  object  as  if  each  projection  was  in  the  same 
plane,  e.g.,  suppose  a  cube,  Fig.  143,  situated  in  space  in  the 
third  angle,  the  point  of  sight  in  front  of  it,  and  the  direction 
of  the  rays  of  light  coinciding  with  the  diagonal  of  the  cube, 
as  shown  by  Fig.  144.  Then  the  edges  avb",  bvc°  will  be  shade 
lines,  because  they  are  the  edges  which  separate  the  illumin- 
ated faces  (the  faces  upon  which  fall  the  rays  of  light)  from 
the  shaded  faces,  as  shown  by  Fig.  144. 

Now  the  source  of  light  being  fixed,  let  the  point  of  sight 
remain  in  the  same  position,  and  conceive  the  object  to  be  re- 
volved through  the  angle  of  90°  about  a  hor.  axis  so  that  a 


ORTHOGRAPHIC  PROJECTION. 


10$ 


plan  at  the  top  of  the  object  is  shown  above  the  elevation,  and 
as  the  projected  rays  of  light  falling  in  the  direction  of  the 
diagonal  of  a  cube  make  angles  of  45°  with  the  hor.,  then  with 
the  use  of  the  45°  triangle  we  can  easily  determine  that  the 
lower  and  right-hand  edges  of  the  plan  as  well  as  of  the  ele- 
vation should  be  shade  lines. 

This  practice  then  will  be  followed  in  this  work,  viz. : 
Shade  lines  shall  be  applied  to  all  projections  of  an  object. 


FIG.  143. 


FIG.  144. 


considering  the  rays  of  light  to  fall  upon  each  of  them,  from 
the  same  direction. 

Shade  lines  should  have  a  width  equal  to  3  times  that  of 
the  other  outlines.  Broken  lines  should  never  be  shade  lines. 

The  outlines  of  surfaces  of  revolution  should  not  be  shade 
lines.  The  shade-lined  figures  which  follow  will  assist  in  il- 
lustrating the  above  principles ;  they  should  be  studied  until 
understood. 


106  MECHANICAL   DRAWING. 

SHADES. 

The  shade  of  an  object  is  that  part  of  the  surface  from 
which  light  is  excluded  by  the  object. 

The  line  of  shade  is  the  line  separating  the  shaded  from 
the  illuminated  part  of  an  object,  and  is  found  where  the  rays 
of  light  are  tangent  to  the  object. 

Brilliant  Points. — "  When  a  ray  of  light  falls  upon  a  sur- 
face which  turns  it  from  its  course  and  gives  it  another  direc- 
tion, the  ray  is  said  to  be  reflected.  The  ray  as  it  falls  upon 
the  surface  is  called  the  incident  ray,  and  after  it  leaves  the 
surface  the  reflected  ray.  The  point  at  which  the  reflection 
takes  places  is  called  the  point  of  incidence. 

"  It  is  ascertained  by  experiment — 

"  (a)  That  the  plane  of  the  incident  and  reflected  rays  is 
always  normal  to  the  surface  at  the  point  of  incidence ; 

"  (&)  That  at  the  point  of  incidence  the  incident  and  re- 
flected rays  make  equal  angles  with  the  tangent  plane  or  normal 
line  to  the  surface. 

"  If  therefore  we  suppose  a  single  luminous  point  and  the 
light  emanating  from  it  to  fall  upon  any  surface  and  to  be  re- 
flected to  the  eye,  the  point  at  which  the  reflection  takes  place 
is  called  the  brilliant  point.  The  brilliant  point  of  a  surface 
is,  then,  the  point  at  which  a  ray  of  light  and  a  line  drawn  to 
the  eye  make  equal  angles  with  the  tangent  plane  or  normal 
line — the  plane  of  the  two  lines  being  normal  to  the  surface." 
— Davies  :  Shades  and  Shadows. 

Considering  the  rays  of  light  to  be  parallel  and  the  point 
of  sight  at  an  infinite  distance,  the  brilliant  point  on  the  sur- 
face of  a  sphere  is  found  as  follows:  Let  A°CV  and  AhO\  Fig. 


ORTHOGRAPHIC  PROJECTION. 


lO/ 


145,  be  a  ray  of  light  and  A°Ah  a  visual  ray.  Bisect  the  angles 
contained  between  the  ray  of  light  and  the  visual  ray  as  fol- 
lows: Revolve  A*C*  about  the  axis  A"  until  it  becomes  parallel 
to  the  hor.  plane  at  AVC1".  At  C°  erect  a  per.  to  intersect 
a  hor.  through  Ch  at  Cf  join  C*Lh  (L  may  be  any  convenient 


FIG.  145. 

point  on^the  line  of  vision),  bisect  the  angle  LhAhClh  with  the 
line  AkLP.  Join  ChLh  and  through  the  point  Z>*,  draw  a  hor. 
cutting  ChLh  at  Z\A,  then  AhDlh  is  the  hor.  projection  of  the 
bisecting  line.  A  plane  drawn  per.  to  this  bisecting  line  and 
tangent  to  'the  sphere  touches  the  surface  at  the  points 
B°B*  where  the  bisecting  lines  pierce  it.  Therefore  B°Bh  are 
the  two  projections  of  the  brilliant  point. 


io8 


MECHANICAL   DRAWING. 


The  point  of  shade  can  be  found  as  follows: 
Draw  AhG,  Fig.   145,  making  an  angle  of  45°  with  a  hor. 
Join  the  points  E  and  /'with  a  straight  line  EF.     Lay  off  on 
AhG  a  distance   equal  to  EF,  and  join  EG.      Parallel  to  EG 
FIG.  146.  FIG.  147. 


FIG.  148. 

draw  a  tangent  to  the  sphere  at  the  point  T.  Through  T 
draw  TPh  per.  to  AhG.  From  the  point  Ph  drop  a  per.  to  P°. 
Pv  is  the  point  of  shade. 

PROB.  24. — To  shade  the  elevation  of  a  sphere  with  graded 
arcs  of  circles. 


ORTHOGRAPHIC  PROJECTION. 


109 


First  find  the  brilliant  point  and  the  point  of  shade,  and 
divide  the  radius  I,  2  into  a  suitable  number  of  equal  parts, 
and  draw  arcs  of  circles  as  shown  by  Fig.  146,  grading  them 
by  moving  the  center  a  short  distance  on  each  side  of  the 
center  of  the  sphere  on  the  line  Bh2  and  varying  the  length  of 
the  radii  to  obtain  a  grade  of  line  that  will  give  a  proper 
shade  to  the  sphere.  It  is  desirable  to  use  a  horn  center  to 
protect  the  center  of  the  figure. 

Fig.    149   shows    the    stippling  method    of    shading    the 
sphere. 


FIG.  149. 


FIG.  150. 


PROB.  25.— To  shade  a  right  cylinder  with  graded  right 
lines. 

Find  the  line  of  light  B°  by  the  same  method  used  to  find 
the  brilliant  point  on  the  sphere,  except  that  the  line  of  light 
is  projected  from  the  point  Bh  where  the  bisection  line  AhD 
cuts  the  circle  of  the  cylinder. 

The  line  of  shade  is  found  where  a  plane  of  rays  is  tan- 
gent to  the  cyl.  at  S°  and  Sh. 

Fig.  150  shows  how  the  shading  lines  are  graded  from 
the  line  of  shade  to  the  line  of  light. 

It  will  be  noticed  that  the  lines  grow  a  little  narrower  to 
the  right  of  the  line  of  shade  on  Fig.  150;  this  shows  where 


no 


MECHANICAL   DRA  WING. 


the  reflection  of  the  rays  of  light  partly  illumine  the  outline 
of  the  cylinder. 

PROB.  26,  FlG.  148. — To  shade  a  right  cone  with  graded 
right  lines  tapering  toward  the  apex  of  the  cone. 

Find  the  elements  of  light  and  shade  as  shown  by  Fig.  148, 
and  draw  the  shading-lines  as  shown  by  Fig.  151,  grading 
their  width  toward  the  light  and  tapering  them  toward  the 
apex  of  the  cone. 


FIG.  151.  FIG.  152. 

The  mixed  appearance  of  the  lines  near  the  apex  of  the 
cone  on  Fig.  151  can  usually  be  avoided  by  letting  each  line 
dry  before  drawing  another  through  it,  or  as  some  draftsmen 
do,  stop  the  lines  just  before  they  touch. 

PROB.   27. — To  shade  the  concave  surface  of  a  section  of  a 

hollow  cylinder. 

Find  the  element  of  light 
and  grade  the  shading  lines 
from  it  to  both  edges  as  shown 
by  Fig.  152. 

FIG.  153.  Fig.    153    shows   a    conven- 

tional method  of  shading  a  hexagonal  nut. 


ORTHOGRAPHIC  PROJECTION. 


Ill 


SHADOWS. 


Let  Rj   Fig.    154,   be  the  direction  of  the    rays  of  light 
and   C  an  opaque  body   between   the  source  of  light  and  a 


FIG.  154. 

surface  5.  The  body  C  will  prevent  the  rays  from  passing 
in  that  direction,  and  its  outline  will  be  projected  at  D  on 
the  surface  5.  D  is  the  shadow  of  C. 

The  line  which  divides  the  illuminated  portion  of  the 
surface  5  from  the  shadow  D  is  called  the  line  of  shadow. 

Shadow  of  a  Point. — If  a  line  is  drawn  through  a  point  in 
space  in  a  direction  opposite  to  the  source  of  light,  the  point 
in  which  this  line  pierces  the  plane  of  projection  is  the 
shadow  of  the  point  on  that  plane. 


112 


ME CHA NICA L   DRA  WING . 


To  find  the  shadow  on  the  H.P.  of  a  point  in  space  in 
the  first  dihedral  angle: 

Let  A,  Fig.  155,  be  the  point  in  space,  and  R  the 
direction  of  the  ray  of  light;  then  A/f  is  the  shadow  of  the 
point  A  on  H.P.,  and  AIfA"  is  the  hor.  proj.  and  ArA^  the 


FIG.  155. 

vert.  proj.  of  A\  Br  is  the  point  where  R  pierces  V  when 
prolonged  below  H.P.,  and  BH  is  its  hor.  proj.  in  the  G.L. 
The  projections  of  R  would  then  be  AVBV  and  AHBH. 

The  shadow  of  a  point  in  }T  may  be  found  in  a  similar 
manner, 

SJiadoivs  of  Rig] it  Lines. — The  shadow  of  a  right  line  on 
a  plane  may  be  determined  by  finding  the  shadows  of  two  of 
its  points  and  joining  these  by  a  right  line;  e.g.,  the  shadow 
of  the  line  AB,  Fig.  156,  on  H.P.  is  found  as  follows: 

Through  the  points  Al'B1'  draw  the  rays  AVA?  and  BrBf 
to  intersect  the  plane  of  projection  in  G.L.  in  the  points  A^ 
and  /?,'';  from  these  points  drop  perpendiculars  to  meet  rays 
drawn  through  A11  and  BH  in  the  points  A*1  and  B".  A  line 
drawn  from  A"  to  B"  is  the  shadow  of  AB  on  H.P. 

If  a  righ  tline  is  parallel  to  the  plane  of  projection  its 
shadow  will  be  parallel  to  the  line  itself. 


ORTHOGRAPHIC   PROJEWION. 


113 


If  a  line  coincides  with  a  ray  of  light,  its  shadow  on  any 
surface  will  be  a  point. 


PROB.   28. — To  find  the  shadow  of  a  right  line  on   V.P. 
and  H.P: 

Let  ABj  Fig.  157,  be  the  given  line.     Find  the  shadows 


FIG.  157. 


114 


ME  CHA  NIC  A  L    DRA  WING . 


of  the  points  A  and  B  by  passing  rays  through  each  of  their 
projections  to  make  angles  of  45°  with  G.L.  The  shadow  of 
Au  on  H.P.  is  found  at  A?,  and  that  of  BH  at  £/7,  where  the 
rays  through  these  points  intersect  the  H.P.  The  shadow 
of  Av  on  V.P.  is  found  at  A,1''  and  that  of  Bv  at  B?,  where 
the  rays  through  these  points  intersect  V.P.  Join  A"  and 
B*1  with  a  straight  line  and  we  have  the  shadow  of  AB  on 
H.P.,  and  the  shadow  on  V.P.  is  found  in  the  same  way  by 
joining  with  a  straight  line  the  points  A*'  and  B? . 

That  part  of  the  shadow  which  falls  on  V.P.  below  G.L., 
and  on  H.P.  above  G.L.,  is  called  the  secondary  shadow, 
because  it  makes  a  second  intersection,  i.e.,  it  is  conceived 
to  have  passed  through  V.P.  and  made  an  intersection  with 
H.P.  behind  V.P.  With  the  use  of  the  secondary  shadow 
problems  like  this  are  easier  of  solution. 


ORTHOGRAPHIC  PROJECTION.  1 15 

PROB.  29. — ABCD,  Fig.  158,  is  a  square  plate  parallel  to 
V.P.  ;  find  its  shadow  on  H.P. 

Through  the  points  Av\  Bv ,  Dl\  and  A"CH,  BH DH ,  draw 
rays  making  the  angle  of  45°  (or  any  other  angle  which  may 
be  adopted)  with  G.L.,  and  determine  the  shadows  of  these 
points  as  explained  in  Fig.  155.  They  will  be  found  in  the 
points  A"B",  C",  D".  Join  these  points  with  right  lines 
and  they  will  form  the  line  of  shadow  of  the  square  plate  on 
H.P. 

PROB.  30. — To  find  the  shadozv  of  a  cube  on  V.P.  with 
one  face  in  V.P.  and  the  other  faces  parallel  or  perpendicular 
to  H.P. 

Fig.  159  shows  the  cube  in  the  given  position.  The  line 
C  A  DvBr 


FIG.  159. 

of  shade  is  composed  of  edges  EF,  FG,  GD,  DB,  and  the 
edges  AE  and  AB  in  V.P.  which  coincide  with  their  shadows. 


MECHANICAL   DRAWING. 

The  shadow  of  EF  is  ErF»  of  FG  is  F\  Glt  of  GD  is  G,D19 
of  Z>.Z?  is  D,BV.  The  shadows  of  the  edges  AE  and  ^4/? 
coincide  with  the  lines.  These  shadows  are  found  by  the 
same  rules  used  for  rinding  the  shadows  of  a  line  in  Prob.  28. 
The  line  of  shadow  is  BrD,Gf,FvEvAvDv.  The  visible  line 
of  shadow  is  BVD1G,F1E1'CVDV . 

PROB.  31. —  To  find  tJie  shadow  of  a  rectangular  abacus  on 
the  face  of  a  rectangular  pillar. 

Assume  the  hor.  and  vert,  projs.  of  the  abacus  and  pillar 
to  be  as  shown  in  Fig.  160. 


-B- 


The  line  of  shade  of  the  abacus  is  seen  to  be  the  edges 
A.UB^  and  A/'C/'.  The  plane  of  rays  through  edge  A^B," 
is  per.  to  V.P.,  and  the  line  AlvEt  is  its  vert.  proj.  or  trace; 
its  hor.  trace  is  A"Eir.  The  shadow  on  the  left  side  face,  is 
vertically  projected  in  the  point  E*r  where  the  plane  of  rays 
intersects  that  face.  The  ray  through  the  point  A"  pierces 
the  front  face  in  the  point  EH ,  which  is  the  shadow  of  A" , 


ORTHOGRAPHIC  PROJECTION. 


117 


and  EfE1*,  E,vev  is  the  shadow  of  the  part  F^A,"  on  this 
face. 

The  line  A"C"  is  parallel  to  the  front  face,  therefore  its 
shadow  on  it  will  be  parallel  to  itself  and  pass  through  E. 

The  visible  line  of  shadow  is  now  found  to  be  I  E^EVHV2  i. 

PROB.  32. — Construct  the  shade  of  an  upright  hex.  prism 
and  its  shadow  on  both  planes. 

Fig.    161    shows  the  given   prism  with  its   line  of   shade 


-A— 


B' 


FIG    161. 


A?B?E,VDVFV  on  the  vert,   pro].,  CHDHFHEH  on  the  hor. 
proj.,  and  its  shadow  on  both  planes. 

PROB.  33. — Given  a  circular  plate  parallel  to  one  coordin- 
ate plane  ;  construct  its  shadoiv  on  the  other  plane. 


MECHANICAL    DRAWING. 

Let  AvBvCvDvm&  AHCH,  Fig.  162,  be  the  projections 
of  the  circular  piate. 

Circumscribe  a  square  EVGV  about  the  circle;  its  shadow 
on  H.P.  will  be  the  parallelogram  AHGH,  and  the  shadows 
of  the  points  AVB1CVDV  are  projected  in  the  points 


Fir,.  162. 

.  The  shadow  of  the  inscribed  circle  is  an  el- 
lipse tangent  to  the  parallelogram  at  the  points  A"B"C"D", 
with  B"D"  and  A"C"  as  conjugate  diameters. 

The  position  and  length  of  the  axes  of  the  ellipse  of 
shadow  may  be  found  as  follows: 

Erect  a  perpendicular  at  the  point  C  making  GrKr  equal 
to  radius  of  the  circle'  draw  KOP\  then  KP  is  equal  to  the 
major  and  MK  to  the  minor  axis,  and  angle  0  is  twice  the 
angle  of  the  transverse  axis  with  the  horizontal  conjugate 
diam. ;  i.e.,  KP  is  equal  to  i,  2,  MK  to  3,  4,  and  2,  Of", 
or  angle  6,  is  equal  to  half  KOC  v. 


ORTHOGRAPHIC  PROJEC7*ION. 

PROB.  34. — Find  the  shade  of  a  cylindrical  column  and 
abacus,  and  the  shadow  of  the  abacus  on  the  column. 

Let  AvBvCr  w&  AHBHCn,  Fig.  163,  be  the  projections 
of  the  abacus,  DHEHFH  and  DHDVGVFH  the  projections  of 
the  column. 


G-A 


FIG.  163. 

The  line  of  shade  on  the  column  is  found  by  passing  two 
planes  of  rays  tangent  to  the  column  perpendicular  to  H.P. 
and  parallel  to  the  hor.  proj.  of  the  ray  of  light.  KL  and 
EH  are  the  traces  of  these  planes  tangent  to  the  column  at 
the  points  Ly  and  EH  and  MN  the  visible  line  of  deepest 
shade  on  the  cylindrical  column. 

The  deepest  line  of  shade  I,  2  on  the  abacus  is  found  in 
the  same  way. 

The  line  of  shadow  on  the  column  of  that  portion  of  the 
lower  circumference  of  the  abacus  which  is  toward  the  source 
of  light  is  found  by  passing  vertical  planes  of  rays,  as  3,4,  to 


UBRAftp 

&€&* 


120 


MECHANICAL    DRAWING. 


determine  any  number  of  points  in  the  line,  and  joining  these 
points  by  a  line  as  shown  in  Fig.  163. 

PROB.     35. — Find  tJie    shade    of  an    oblique   cone   and  its 
shadow  on  H.P. 


Take  the  cone  as  given  in  Fig.  164.  Pass  two  planes  of 
rays  tangent  to  the  cone;  their  elements  of  contact  will  be 
the  deepest  lines  of  shade.  To  determine  the  elements  of 
contact  draw  a  ray  through  Cv\  C"  i«s  its  hor.  trace.  From 


OR  THOGRA PHIC  PROJECTION. 


121 


C"  draw  lines  tangent  to  the  base  at  D  and  £•  the  lines  of 
contact  are  CE  and  CD,  and  ECD  is  the  line  of  shade. 

The  visible  line  of  shade  on  H.P.  is  EHDH,  and  on  V.P. 
it  is  CVEV.  The  shadow  on  H.P.  is  EHC*{DH. 

PROB.  36. — To  draw  a  front  and  end  elevation  of  a  rect- 
angular hollow  box  with  a  rectangular  block  on  each  face,  each 
block  to  have  a  rectangular  opening,  and  all  to  be  properly 
shade-lined  and  drawn  to  the  dimensions  given  on  Fig.  165. 

Draw  the  hor.  center  line  first,  and  then  the  vertical  center 
line  of  the  end  view.  About  these  center  lines  on  the  end  el- 

FIG.  165. 


FIG.  166. 

evation  construct  the  squares  shown  and  erect  the  edges  of  the 
blocks.      Next  draw  the  hidden  lines  indicating  the  thickness 


122  MECHANICAL   DRAWING. 

of  the  walls  of  the  box  and  the  openings  through  the  blocks, 
measuring  the  sizes  carefully  to  the  given  dimensions. 

Draw  the  front  elevation  by  projecting  lines  from  the  va- 
rious points  on  the  end  elevation,  and  assuming  the  position  of 
the  line  AB  measure  off  the  lengths  of  the  hor.  lines  and  erect 
their  vert,  boundaries  as  shown  by  the  figure. 

PROB.  37. — Given  the  end  elevation  of  the  last  prob.,  cut 
by  three  planes  A,  B  and  C,  Fig.  166.  Draw  the  projections 
of  these  sections  when  the  part  to  the  left  of  the  cutting  plane 
has  been  removed,  and  what  remains  is  viewed  in  the  direction 
of  the  arrow,  remembering  that  all  the  visual  rays  are  parallel. 

These  drawings  and  all  that  may  follow  are  to  be  properly 
shade-lined  in  accordance  with  the  principles  given  above. 

ISOMETRICAL   DRAWING. 

In  orthographic  projection  it  is  necessary  to  a  correct 
understanding  of  an  object  to  have  at  least  two  views,  a  front 
and  end  elevation,  or  an  elevation  and  plan,  and  sometimes 
even  three  views  are  required. 

Isometric  drawing  on  the  other  hand  shows  an  object  com- 
pletely with  only  one  view.  It  is  a  very  convenient  system 
for  the  workshop.  Davidson  in  his  Projection  calls  it  the 
"  Perspective  of  the  Workshop."  It  is  more  useful  than  per- 
spective for  a  working  drawing,  because,  as  its  name  implies 
("  equal  measures  ")  it  can  be  made  to  any  scale  and  measured 
like  an  orthographic  drawing.  It  is,  however,  mainly  em- 
ployed to  represent  small  objects,  or  large  objects  drawn  to  a 
small  scale,  whose  main  lines  are  at  right  angles  to  each  other. 

The  principles  of  isometrical  drawing  are  founded  on  a 
cube  resting  on  its  lower  front  corner,  I,  Fig.  167,  and  its  base 


ORTHOGRAPHIC  PROJECTION. 


123 


elevated  so  that  its  diagonal  AB  is  parallel  to  the  horizontal 
plane.  Then  if  the  cube  is  rotated  on  the  corner  I  until  the 
diagonal  AB  is  at  right  angles  to  the  vert,  plane,  i.e., 
through  an  angle  of  90°,  the  front  elevation  will  appear  as 
shown  at  i,  2,  3,  4,  Fig.  167,  a  regular  hexagon. 

Now  we  know  that  in  a  regular  hexagon,  as  shown  by  Fig. 
167,  the  lines  lA,  A$,  etc.,  are  all  equal,  and  are  easily  drawn 


FIG.  167. 

with  the  30°  X  60°  triangle.  But  although  these  lines  and 
faces  appear  to  be  equal,  yet,  being  inclined  to  the  plane  of 
projection,  they  are  shorter  than  they  would  actually  be  on 
the  cube  itself.  However,  since  they  all  bear  the  same  pro- 
portion to  the  original  sizes,  they  can  all  be  measured  with 
the  same  scale. 

We  will  now  describe  the  method  of  making  an  isomet- 
rical  scale. 

Draw  the  half  of  a  square  with  sides  —  2^",  Fig.  168. 
These  two  sides  will  make  the  angle  of  45°  with  the  horizontal. 
Now  the  sides  of  the  corresponding  isometrical  square,  we  have 
seen,  make  the  angle  of  30°  with  the  horizontal,  so  we  will 


124 


MECHANICAL    DRA  WING. 


draw  14,  34,  making  angles  of  30°  with  1,3.  The  differ- 
ence then  between  the  angle  2,  I,  3  and  the  angle  4,  I,  3  is 
15°,  and  the  proportion  of  the  isometrical  projection  to  the 
actual  object  is  as  the  length  of  the  line  3,  2  to  the  line  3,  4. 
And  if  the  line  3,  2  be  divided  into  any  number  of  equal  parts, 
and  lines  be  drawn  through  these  divisions  par.  to  2,  4  to  cut 
the  line  3,  4  in  corresponding  divisions,  these  will  divide  3,  4 
proportionately  to  3,  2. 

Now  if  the  divisions  on  3,  2  be  taken  to  represent  feet 
and  those  on  3,  4  to  represent  2  feet,  then  3,  4  would  be  aa 
isometrical  scale  of  . 


FIG.  168. 

Since  isometrical  drawings  may  be  made  to  any  scale,  we 
may  make  the  isometrical  lines  of  the  object  =  their  true  size. 
This  is  a  common  practice  and  precludes  the  need  of  a  special 
isometrical  scale. 

The  Direction  of  the  Rays  of  Light. — The  projection  of  a 
ray  of  light  in  isometrical  drawing  will  make  the  angle  of  30° 
with  the  horizontal  as  shown  by  the  line  3,  2  on  the  front 
elevation  of  the  hex.,  Fig.  167.  And  the  shade  lines  will  be 
applied  as  in  ordinary  projection. 

PROB.  38. — To  make  the  isometrical  drawing  of  a  two- 
armed  cross  standing  on  a  square  pedestal. 


ORTHOGRAPHIC  PROJECTION. 


125 


Begin  by  drawing  a  center  line  AB,  Fig.  169,  and  from  the 
point  A  draw  AC  and  AD,  making  an  angle  of  30°  with  the 
horizontal.  Measure  from  A  on  the  center  line  AB  a  dis- 
tance -~  TV',  and  draw  lines  par.  to  AC,  AD',  make  AC  and 
AD  2f"  long  and  erect  a  perpendicular  at  D  and  C,  complet- 
ing the  two  front  sides  of  the  base,  etc. 


FIG.  169. 


PROB.  39. — To  make  the  isometrical  drawing  of  a  hollow 
cube,  with  square  block  on  each  face  and  a  square  hole 
through  each  block,  to  dimensions  given  on  Fig.  170. 

As  before,  first  draw  a  center  line,  and  make  an  isometrical 
drawing  of  a  2f"  cube,  and  upon  each  face  of  it  build  the 
blocks  with  the  square  holes  in  them,  exactly  as  shown  in 
Fig.  170. 

PROB.  40. — To  project  an  isometrical  circle. 

The  circle  is  enclosed  in  a  square,  33  shown  by  Fig.  171. 


126 


MECHANICAL    DRA  WING. 


Draw  the  circle  with  a  radius  =  2"  and  describe  the  square 
I,  2,  3,  4  about  it. 

Draw  the  diagonals  i,  2,  3,  4  and  the  diameters  5,  6,  7,  8 
at  right  angles  to  each  other. 

Now  from  the  points  i  and  2  draw  lines  lA,  iB  and  2 A, 
2B,  making  angles  of  30°  with  the  hor.  diagonal  1,2.  And 


FIG.  170. 

through  the  center  O  draw  CD  and  EF  at  right  angles  to  the 
isometrical  square. 

The  points  CD,  EF,  and  GH  will  be  points  in  the  curve 
of  the  projected  isometrical  circle,  which  will  be  an  ellipse. 
The  ellipse  may  be  drawn  sufficiently  accurate  as  follows: 
With  center  B  and  radius  BC  describe  the  arc  CF  and  ex- 
tend it  a  little  beyond  the  points  C  and  F,  and  with  center  A 
and  same  rad.  describe  a  similar  arc,  then  with  a  rad.  which 


ORTHOGRAPHIC  PROJECTION, 


127 


FIG.  171. 


FIG.  172. 


FIG.  173- 


FIG.  174- 


FIG.  175- 


FIG.  176. 


1 2  8  ME  CHA  NIC  A  L   DRA  WING . 

may  readily  be  found  by  trial,  draw  arcs  through  the  points 
G  and  //and  tangent  to  the  two  arcs  already  described. 

Figs.  172,  173,  174,  175,  176,  and  177  are  for  practice  in 
the  application  of  the  preceding  principles,  and  at  least  one 
of  these  should  be  drawn,  or  it  would  be  better  still  if  the  stu- 


FIG.  177. 

dent  would  attempt  to  make  an  isometrical  projection  of  his 
instrument-box,  desk,  or  any  familiar  object  at  hand.  These 
figures  may  be  measured  with  the  i£"  scale  and  drawn  with 
the  2"  scale. 

WORKING   DRAWINGS. 

Working  drawings  are  sometimes  made  on  brown  detail- 
paper  in  pencil,  traced  on  tracing-paper  or  cloth,  and  then 
blue  printed. 

The  latter  process  is  accomplished  as  follows: 

The  tracing  is  placed  face  down  on  the  glass  in  the  print- 
ing-frame, and  the  prepared  paper  is  placed  behind  it,  with 
the  sensitized  surface  in  contact  with  the  back  of  the  tracing. 

In  printing  from  a  negative  the  sensitized  surface  of  the 
prepared  paper  is  placed  in  contact  with  the  film  side  of  the 
negative,  and  the  face  is  exposed  to  the  light. 

The  blue-print  system  for  working  drawings  has  many 
drawbacks,  e.g.,  the  sectioned  parts  of  the  drawing  require  to 


ORTHOGRAPHIC  PROJECTION.  1 29 

be  hatch-lined,  using  the  standard  conventions  already  re 
ferred  to  for  the  different  materials.  This  takes  a  great  deal 
of  time.  The  print  has  usually  to  be  mounted  on  cardboard, 
although  this  is  not  always  done,  and  unless  it  is  varnished 
the  frequent  handling  with  dirty,  oily  fingers  soon  makes  it 
unfit  for  use. 

Changes  can  be  made  on  the  prints  with  soda-water,  it  is 
true,  but  they  seldom  look  well,  and  when  many  changes  or 
additions  require  to  be  made  it  is  best  to  make  them  on  the 
tracing  and  take  a  new  print.  And  the  sunlight  is  not  always 
favorable  to  quick  printing.  So  taking  everything  into  con- 
sideration the  system  of  making  working  drawings  directly  on 
cards  and  varnishing  them  is  probably  the  best.  It  is  the 
system  used  by  the  Schenectady  Locomotive  Works  and 
many  other  large  engineering  establishments.  In  size  the 
cards  are  made  9"  X  12",  12"  X  18",  18"  X  24";  they  are 
made  of  thick  pasteboard  mounted  with  Irish  linen  record- 
paper.  The  drawings  are  pencilled  and  inked  on  these  cards 
in  the  usual  way,  and  the  sections  are  tinted  with  the  conven- 
tional colors,  which  are  much  quicker  applied  than  hatch- 
lines.  The  face  of  the  drawing  is  protected  with  two  coats  of 
white  shellac  varnish,  while  the  back  of  the  card  is  usually 
given  a  coat  of  orange  shellac. 

The  white  varnish  can  easily  be  removed  with  a  little 
alcohol,  and  changes  made  on  the  drawing,  and  when  revar- 
nished  it  is  again  ready  for  the  shop. 

We  will  now  try  to  describe  what  a  working  drawing  is 
and  what  it  is  for.  In  the  hands  of  an  experienced  workman 
a  working  drawing  is  intended  to  convey  to  him  all  the  neces- 
sary information  as  to  shape,  size,  material,  and  finish  to 


130 


MECHANICAL    DRA  WING. 


ORl^HOGRAPHIC  PROJECTION. 


132  MECHANICAL   DRAWING. 

enable  him  to  properly  construct  it  without  any  additional  in- 
structions. This  means  that  it  must  have  a  sufficient  num- 
ber of  elevations,  sections,  and  plans  to  thoroughly  explain 
and  describe  the  object  in  every  particular.  And  these  views 
should  be  completely  and  conveniently  dimensioned.  The 
dimensions  on  the  drawing  must  of  course  give  the  sizes  to 
which  the  object  is  to  be  made,  without  reference  to  the  scale 
to  which  it  may  be  drawn.  The  title  of  a  working  drawing 
should  be  as  brief  as  possible,  and  not  very  large — a  neat, 
plain,  free-hand  printed  letter  is  best  for  this  purpose. 

Finished  parts  are  usually  indicated  by  the  letter  "  f,"  and 
if  it  is  all  to  be  finished,  then  below  the  title  it  is  customary 
to  write  or  print  "  finished  all  over." 

The  number  of  the  drawing  may  be  placed  at  the  upper 
left-hand  corner,  and  the  initials  of  the  draftsman  immedi- 
ately below  it. 

A  second-year  course,  entitled  Mechanical  Drawing  and 
Elementary  Machine  Design,  is  in  preparation,  and  will  shortly 
be  published. 

Figs.  178  and  179  show  working  drawings  of  two  shaft- 
couplings,  fully  figured,  sectioned,  and  shade-lined. 


INDEX. 


A 

PAGE 

Angle,  To  bisect  an 19 

Angle,  To  construct  an 15 

Anti-friction  curve,  "  Schiele's" 50 

Arched  window-opening,  To  draw  an 53 

Arkansas  oil-stones 5 


B 

Baluster,  To  draw  a 53 

Board,  Drawing I 

Bow  instruments 2 

Brass,  Sheet  of 6 

Breaks,  Conventional 61 

Brilliant  points 106 


C 

Celluloid,  Sheet  of  thin 5 

Center  lines 60 

Cinquefoil  ornament,  To  draw  the 53 

Circle,  Arc  of  a.  To  find  the  center  of  an , 32 

Circle,  Arc  of  a,  To  draw  a  line  tangent  to  an 33 

Circle,  To  draw  a  right  line  equal  to  half  the  circumference  of  a 31 

Circle,  To  draw  a  tangent  between  two 33 

Circle,  To  draw  tangents  to  two 34 

Circle,  To  draw  an  arc  of  a,  tangent  to  two  straight  lines 34 

Circle,  To  inscribe  a,  within  a  triangle 35 

Circle,  To  draw  an  arc  of  a,  tangent  to  two  circles 36 

Circle,  To  draw  an  arc  of  a,  tangent  to  a  straight  line  and  a  circle 37 

Circle,  To  construct  the  involute  of  a <  45 

Circle,  To  find  the  length  of  an  arc  of  a,  approximately 47 

133 


1 34  INDEX. 

PAGE 

Cissoid,  To  draw  the - 49 

Compass .......  2 

Conventions 56 

Conventions,  Shading 104 

Conventional  breaks 61 

Conventional  lines 60 

Conventional  screw-threads 62 

Cross-sections 62 

Curves,  Irregular   3 

Cycloid,  To  describe  the 46 


D 

Dark  surfaces 104 

Development  of  the  surfaces  of  a  hexagonal  prism go 

Development  of  the  surface  of  a  right  cylinder g2 

Development  of  the  surface  of  a  cone 93 

Development  of  the  surface  of  a  cylindrical  dome 96 

Development  of  a  locomotive  gusset  sheet 97 

Dihedral  angles 75 

Direction,  The,  of  the  rays  of  light 105 

Dividers,   Hair-spring 2 

Drawing-board I 

Drawing-pen ...  2 

Drawing  to  scale   12,  54 


E 

Ellipse,  To  describe  an 38 

Ellipse,  Given  an,  to  find  the  axes  and  foci 43 

Epicycloid,  To  describe  the 48 

Epicycloid,  To  describe  an  interior 50 

Equilateral  triangle,  To  construct  an 24 

Examples  of  working  drawings 120 


F 

Figuring  and  lettering 66 

Finished  parts  of  working  drawings 122 

G 

Geometrical  drawing 16 

Glass-paper  pencil  sharpener 4 

Gothic  letters 69 


INDEX.  135 

H 

PAGE 

Heptagon,  To  construct  a 28 

Hyperbola,  To  draw  an   42 

Hypocycloid,  To  describe  the .     48 


I 

Ink  eraser 4 

Inks 4 

Instruments   2 

Intersection,  The,  of  a  cylinder  with  a  cone 93 

Intersection,  The,  of  two  cylinders 96 

Intersection,  The,  of  a  plane  with  an  irregular  surface  of  revolution 102 

Involute,  of  a  circle,  To  construct  the 45 

Isometrical  cube 113 

Isometrical  drawing 1 12 

Isometrical  drawing,  Direction  of  the  rays  of  light  in. .*. . .  114 

Isometrical  drawing  of  a  two-armed  cross 115 

Isometrical  drawing  of  a  hollow  cube 1 16 

Isometrical  drawing,  Examples  of '. . .  117 

Isometrical  scale,  The 114 


L 

Leads  for  compass , 13 

Lettering  and  figuring 64 

Line  of  shade 106 

Line,  To  draw  a,  parallel  to  another 19 

Line,  To  divide  a 21 

Line  of  motion 60 

Line  of  section   60 


M 

Mechanical  drawing  and  elementary  machine  design 122 

Model  of  the  co-ordinate  planes   8 1 

Moulding,  The  "Scotia  " 51 

Moulding,  The  "  Cyma  Recta" 51 

Moulding,  The  "  Cavetto  "  or  "  Hollow  " 51 

Moulding,  The  "  Echinus,  "  "  Quatrefoil,"  or  "  Ovolo" 52 

Moulding,  The  "Apophygee" 52 

Moulding,  The  "  Cyma  Reversa  "   52 

Moulding,  The  ' '  Torus  " 52 


136  INDEX. 

N 
Notation 80 


PACK 

Needles 6 


O 

Octagon,  To  construct  an 28 

Orthographic  projection ...      74 

Oval,  To  construct  an 43 


P 

Paper 2 

Parabola,  To  construct  a 41 

Pencil 2 

Pencil  eraser   4 

Pencil,  To  sharpen  the 8 

Pen,  Drawing 9 

Pen,  To  sharpen  the  drawing. 10 

Pentagon,  To  construct  a 28 

Perpendicular,  To  erect  a. . .  . , '. .  17 

Planes  of  projection,  The   75 

Polygon,  To  construct  a 26 

Projection,  The,  of  straight  lines 82 

Projection,  The,  of  plane  surfaces 84 

Projection,  The,  of  solids 90 

Projection,  The,  of  the  cone 93 

Projection  of  the  helix  as  applied  to  screw-threads 99 

Proportional,  To  find  a  mean,  to  two  given  lines 31 

Proportional,  To  find  a  third,  to  two  given  lines 31 

Proportional,  To  find  a  fourth,  to  three  given  lines 32 

Protractor 6 


0 

Quatrefoil,  To  draw  the 53 


R 

Rays  of  light 104 

Rays,  Visual , 104 

Rhomboid,  To  construct  the 21 

Right  angle,  To  trisect  a 24 

Roman  letters 57 


INDEX.  137 


M6B 

Scale  guard 6 

Scale,  Drawing  to 12,  54 

Scale,  To  construct  a 55 

Schiele's  curve,  To  draw 50 

Screw-threads,  Conventional 62 

Screw-threads,  Regular 100 

Section  lines , 56 

Section  lines,  Standard 5& 

Shade  lines  and  shading 103 

Shade,  To,  the  elevation  of  a  sphere < 108 

Shade,  To,  a  right  cylinder 109 

Shade,  To,  a  right  cone no 

Shade,  To,  a  concave  cylindrical  surface   no 

Shadows Ill 

Sharpen  pencil,  To 8 

Sharpen  pen,  To 10- 
Sheet  brass 6 

Sheet  celluloid 6 

"  Sibley  College  "  set  of  irregular  curves 3 

"  Sibley  College  "  set  of  instruments 2 

Source  of  light 104 

Spiral,  To  describe  the 44 

Sponge  rubber 5 

Square,  To  construct  a 25 

Stippling 109 

T 

Tacks 5 

T-square 2 

Third  dihedral  angle 75 

Tinting  brush 5 

Tinting  saucer 5 

Title,  The,  of  a  working  drawing 122 

Tracing-ck>th. 6 

Trefoil,  To  describe  the 53 

Triangles 3 

Triangle,  To  construct  a 25 

Triangular  scale 3 

Type  specimens 70 

U 

Use  of  instruments 7 

Use  of  pencil 8 


138  INDEX. 

PAGE 

Use  of,  drawing-pen g 

Use  of  triangles   n 

Use  of  T-square !  r 

Use  of  drawing- board u 

Use  of  scale 12 

Use  of  compasses   13 

Use  of  dividers  or  spacers   13 

Use  of  spring  bows 14 

Use  of  irregular  curves 14 

Use  of  protractor 14 

V 

Visual  rays 104 

Volute,  To  describe  the  "  Ionic  " 45 

W 

Water-colors 5 

Water  glass    5 

Writing- pen 6 

Working  drawings 118 

Working  drawings,  Method  of  making 119 

Working  drawing,  What  is  a 119 

Working  drawings,  Examples  of 119 


J- 


^=====f======^- 


11 


oer  so  j 

4Nlar'5QJ6 


5  1997, 


